Recent zbMATH articles in MSC 15https://zbmath.org/atom/cc/152021-11-25T18:46:10.358925ZWerkzeugEigenvector visualization and arthttps://zbmath.org/1472.000492021-11-25T18:46:10.358925Z"Griffith, Daniel A."https://zbmath.org/authors/?q=ai:griffith.daniel-aSummary: Existing interfaces between mathematics and art, and geography and art, began overlapping in recent years. This newer overarching intersection partly is attributable to the scientific visualization of the concept of an eigenvector from the subdiscipline of matrix algebra. Spectral geometry and signal processing expanded this overlap. Today, novel applications of the statistical Moran eigenvector spatial filtering (MESF) methodology to paintings accentuates and exploits spatial autocorrelation as a fundamental element of art, further expanding this overlap. This paper studies MESF visualizations by compositing identified relevant spatial autocorrelation components, examining a particular Van Gogh painting for the first time, and more intensely re-examining several paintings already evaluated with MESF techniques. Findings include: painting replications solely based upon their spatial autocorrelation components as captured and visualized by certain eigenvectors are visibly indistinguishable from their original counterparts; and, spatial autocorrelation supplies measurements allowing a differentiation of paintings, a potentially valuable discovery for art history.On ABC Estrada index of graphshttps://zbmath.org/1472.050392021-11-25T18:46:10.358925Z"Li, Shuchao"https://zbmath.org/authors/?q=ai:li.shuchao"Wang, Lu"https://zbmath.org/authors/?q=ai:wang.lu.4|wang.lu.2|wang.lu.3|wang.lu.1|wang.lu"Zhang, Huihui"https://zbmath.org/authors/?q=ai:zhang.huihuiSummary: Let \(G\) be a graph with vertex set \(V_G = \{ v_1, v_2, \ldots, v_n \}\) and edge set \(E_G\), and let \(d_i\) be the degree of the vertex \(v_i\). The ABC matrix of \(G\) has the value \(\sqrt{ ( d_i + d_j - 2 ) / ( d_i d_j )}\) if \(v_i v_j \in E_G\), and 0 otherwise, as its \((i, j)\)-entry. Let \(\gamma_1, \gamma_2, \ldots, \gamma_n\) be the eigenvalues of the ABC matrix of \(G\) in a non-increasing order. Then the ABC Estrada index of \(G\) is defined as \(\operatorname{EE}_{\mathrm{ABC}}(G) = \sum_{i = 1}^n e^{\gamma_i}\) and the ABC energy of \(G\) is defined as \(\operatorname{E}_{\mathrm{ABC}}(G) = \sum_{i = 1}^n | \gamma_i |\). In this paper, some explicit bounds for the ABC Estrada index of graphs concerning the number of vertices, the number of edges, the maximum degree and the minimum degree, are established. Moreover, some bounds for the ABC Estrada index involving the ABC energy of graphs are also presented. All the corresponding extremal graphs are characterized respectively.Noncorona graphs with strong anti-reciprocal eigenvalue propertyhttps://zbmath.org/1472.050912021-11-25T18:46:10.358925Z"Ahmad, Uzma"https://zbmath.org/authors/?q=ai:ahmad.uzma"Hameed, Saira"https://zbmath.org/authors/?q=ai:hameed.saira"Jabeen, Shaista"https://zbmath.org/authors/?q=ai:jabeen.shaistaSummary: Let \(G\) be a graph having a unique perfect matching and \(A(G)\) be the adjacency matrix of \(G. \, G\) is said to have the strong anti-reciprocal eigenvalue property (property (-SR)) if for each eigenvalue \(\lambda\) of \(A(G)\), its reciprocal \(-1/ \lambda\) is also an eigenvalue of \(A(G)\), with the same multiplicity. In this article, seven classes of noncorona graphs with property (-SR) are obtained.Brouwer type conjecture for the eigenvalues of distance signless Laplacian matrix of a graphhttps://zbmath.org/1472.050922021-11-25T18:46:10.358925Z"Alhevaz, A."https://zbmath.org/authors/?q=ai:alhevaz.abdollah"Baghipur, M."https://zbmath.org/authors/?q=ai:baghipur.maryam"Ganie, Hilal A."https://zbmath.org/authors/?q=ai:ganie.hilal-ahmad"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddinSummary: Let \(G\) be a simple connected graph with \(n\) vertices, \(m\) edges and having distance signless Laplacian eigenvalues \(\rho_1 \geq \rho_2 \geq \dots \geq \rho_n \geq 0\). For \(1 \leq k \leq n\), let \(M_k (G) = \sum^k_{i=1} \rho_i\) and \(N_k (G) = \sum^{k-1}_{i=0} \rho_{n-i}\) be respectively the sum of \(k\)-largest distance signless Laplacian eigenvalues and the sum of \(k\)-smallest distance signless Laplacian eigenvalues of \(G\). In this paper, we obtain the bounds for \(M_k (G)\) and \(N_k (G)\) in terms of the number of vertices \(n\) and the transmission \(\sigma (G)\) of the graph \(G\). We propose a Brouwer-type conjecture for \(M_k (G)\) and show that it holds for graphs of diameter one and graphs of diameter two for all \(k\). As a consequence, we observe that the conjecture holds for threshold graphs and split graphs (of diameter two). We also show that it holds for \(k=n-1\) and \(n\) for all graphs and for some \(k\) for \(r\)-transmission regular graphs.Principal eigenvectors of general hypergraphshttps://zbmath.org/1472.050942021-11-25T18:46:10.358925Z"Cardoso, Kauê"https://zbmath.org/authors/?q=ai:cardoso.kaue"Trevisan, Vilmar"https://zbmath.org/authors/?q=ai:trevisan.vilmarSummary: In this paper, we obtain bounds for the extreme entries of the principal eigenvector of a hypergraph; these bounds are computed using the spectral radius and some classical parameters such as maximum and minimum degree. We also study inequalities involving the ratio and difference between the two extreme entries of this vector.Computation of general Randić polynomial and general Randić energy of some graphshttps://zbmath.org/1472.050992021-11-25T18:46:10.358925Z"Ramane, Harishchandra S."https://zbmath.org/authors/?q=ai:ramane.harishchandra-s"Gudodagi, Gouramma A."https://zbmath.org/authors/?q=ai:gudodagi.gouramma-aThe general Randić matrix of a graph \(G\) of order \(n\), denoted by \(\operatorname{GR}(G)\), is the \(n \times n\) matrix whose \((i, j)\)-entry is \((d_{i}d_{j})^{\alpha}\), \(\alpha \in \mathbb{R}\), if the vertices \(v_i\) and \(v_j\) are adjacent and \(0\) otherwise, where \(d_{i}\) is the degree of \(v_{i}\) for \(i = 1, \ldots, n\). The general Randić polynomial of \(G\) is simply the characteristic polynomial of \(\operatorname{GR}(G)\), and the general Randić energy \(\operatorname{EGR}(G)\) of \(G\) is defined as the sum of the absolute values of the eigenvalues of \(\operatorname{GR}(G)\). In the particular case when \(\alpha = -\frac{1}{2}\), \(\operatorname{EGR}(G)\) is the usual Randić energy of \(G\). In this paper, the authors compute the general Randić polynomial and the general Randić energy of paths, cycles, complete graphs, complete bipartite graphs, friendship graphs and Dutch windmill graphs.A conjecture on the eigenvalues of threshold graphshttps://zbmath.org/1472.051002021-11-25T18:46:10.358925Z"Tura, Fernando C."https://zbmath.org/authors/?q=ai:tura.fernando-colmanA simple graph \(G = (V, E)\) is called a threshold graph if there exists a function \(w: V \rightarrow [0,\infty)\) and a non-negative real number \(t\) such that \(uv \in E\) if and only if \(w(u) + w(v) \ge t\). A simple graph is called an anti-regular graph if only two vertices of it have equal degrees. It is known that, up to isomorphism, there is exactly one connected anti-regular graph on \(n\) vertices, which is denoted as \(A_n\). Anti-regular graphs form a subfamily of the family of threshold graphs. In [\textit{C. O. Aguilar} et al., Linear Algebra Appl. 557, 84--104 (2018; Zbl 1396.05064)], it was conjectured that for each \(n\), \(A_n\) has the smallest positive eigenvalue and the largest negative eigenvalue less than \(-1\) among all threshold graphs on \(n\) vertices. In [\textit{C. O. Aguilar} et al., ibid. 588, 210--223 (2020; Zbl 1437.05128)], this conjecture was proved for all threshold graphs on \(n\) vertices except for \(n - 2\) critical cases where the interlacing method fails. In this paper, the author deals with these cases and thereby completes the proof of this conjecture.Idempotent systems and character algebrashttps://zbmath.org/1472.051602021-11-25T18:46:10.358925Z"Nomura, Kazumasa"https://zbmath.org/authors/?q=ai:nomura.kazumasa"Terwilliger, Paul"https://zbmath.org/authors/?q=ai:terwilliger.paul-mSummary: We recently introduced the notion of an idempotent system. This linear algebraic object is motivated by the structure of an association scheme. There is a type of idempotent system, said to be symmetric. In the present paper we classify up to isomorphism the idempotent systems and the symmetric idempotent systems. We also describe how symmetric idempotent systems are related to character algebras.On distance Laplacian spectrum of zero divisor graphs of the ring \(\mathbb{Z}_n \)https://zbmath.org/1472.130142021-11-25T18:46:10.358925Z"Pirzada, S."https://zbmath.org/authors/?q=ai:pirzada.shariefuddin"Rather, B. A."https://zbmath.org/authors/?q=ai:rather.bilal-a"Chishti, T. A."https://zbmath.org/authors/?q=ai:chishti.tariq-aLet \(R\) be a finite commutative ring with identity \(1\neq 0.\) The zero-divisor graph \(\Gamma(R)\) is the simple undirected graph with the set of all non-zero zero-divisors \(Z^*(R)\) as the vertex set and two distinct vertices \(x\) and \(y\) are adjacent if \(x.y=0\) in \(R.\) Zero-divisor graphs of commutative rings are well studied in several aspects of graph theory for the past three decades. For a graph \(G,\) let \(A(G)\) be the adjacency matrix \(G.\) Let \(Deg(G)\) be the diagonal matrix of vertex degrees of vertices in \(G.\) The matrices \(L(G) = Deg(G) - A(G)\) and \(Q(G) = Deg(G) + A(G)\) are respectively the Laplacian and the signless Laplacian matrices and these matrices are real symmetric and positive semi-definite. It is assumed that \(0=\lambda_n\leq \lambda_{n-1}\leq\cdots\leq \lambda_1\) are the Laplacian eigenvalues of \(L(G).\) In this paper, authors obtained the distance Laplacian spectrum of the zero divisor graphs \(\Gamma(\mathbb{Z}_n\)) for different values of \(n\in \{pq, p^2q, (pq)^2, p^z~\text{for some}~ z\geq 2\}\) where \(p\) and \(q(p < q)\) are distinct primes. Further it is proved that the zero-divisor graph \(\Gamma(\mathbb{Z}_n)\) is distance Laplacian integral if and only if \(n\) is prime power or product of two distinct primes.Green-Lazarsfeld condition for toric edge ideals of bipartite graphshttps://zbmath.org/1472.130242021-11-25T18:46:10.358925Z"Greif, Zachary"https://zbmath.org/authors/?q=ai:greif.zachary"McCullough, Jason"https://zbmath.org/authors/?q=ai:mccullough.jasonThe authors consider toric ideals of bipartite graphs and study the Green-Lazarsfeld condition \(\mathbf{N}_p\), for \(p\geq1\). Their characterizations are given in terms of the combinatorics of the given graph or its complementary graph. They also consider the graded Betti numbers of the toric ideal and give necessary sufficient conditions for not vanishing these numbers.Systems of polynomial equations, higher-order tensor decompositions, and multidimensional harmonic retrieval: a unifying framework. Part II: The block term decompositionhttps://zbmath.org/1472.130472021-11-25T18:46:10.358925Z"Vanderstukken, Jeroen"https://zbmath.org/authors/?q=ai:vanderstukken.jeroen"Kürschner, Patrick"https://zbmath.org/authors/?q=ai:kurschner.patrick"Domanov, Ignat"https://zbmath.org/authors/?q=ai:domanov.ignat"De Lathauwer, Lieven"https://zbmath.org/authors/?q=ai:de-lathauwer.lievenAll secant varieties of the Chow variety are nondefective for cubics and quaternary formshttps://zbmath.org/1472.140082021-11-25T18:46:10.358925Z"Torrance, Douglas A."https://zbmath.org/authors/?q=ai:torrance.douglas-a"Vannieuwenhoven, Nick"https://zbmath.org/authors/?q=ai:vannieuwenhoven.nickLet \(f\in S^d \mathbb C^{n+1}\) be a homogeneous polynomial of degree \(d\) in \(n+1\) variables. The Chow rank of \(f\) is the minimal integer \(s\) such that \(f\) may be written as
\[
f = \ell_{1,1}\cdots \ell_{1,d} + \cdots + \ell_{s,1}\cdots \ell_{s,d},
\]
where the \(\ell_{i,j}\) are linear forms. This is an important instance of an additive decomposition for a tensor. Tensor decompositions are by now a large field with deep geometric and algebraic roots and yet possess a vast number of applications in many contexts such as complexity, information theory, and machine learning among others.
One geometric feature of the subject arises when one asks what is the Chow rank of a generic \(f\in S^d \mathbb C^{n+1}\). Let \(\mathcal{C}_{d,n}\subset \mathbb P^{\binom{n+d}{d}-1}\) be the projective variety parameterizing products of linear forms in \(S^d \mathbb C^{n+1}\). The variety \(\mathcal{C}_{d,n}\) is called the \textit{Chow variety}. Computing the Chow rank of a generic \(f\in S^d \mathbb C^{n+1}\) is equivalent to finding the smallest \(s\) such that \(\sigma_s(\mathcal{C}_{d,n}) = \mathbb P^{\binom{n+d}{d}-1}\), where \(\sigma_s(\mathcal{C}_{d,n})\) is the \(s\)-th secant variety of the Chow variety. The topic of secant varieties is a delightful chapter of classical algebraic geometry that has attracted more attention in the last decades, partly because of its natural role in additive decompositions and applications thereof.
This nice paper is a contribution to determining dimensions of secants of Chow varieties. The main result is that all secant varieties \(\sigma_s(\mathcal{C}_{d,n})\) have expected dimensions for:
\begin{itemize}
\item any \(n\) and \(d=3\),
\item \(n=3\) and any \(d\).
\end{itemize}
The methods are very combinatorial and rely on a lattice construction generalising a method due to Brambilla and Ottaviani. The base cases of the inductions are treated with a computer-assisted proof.Critical loci in computer vision and matrices dropping rank in codimension onehttps://zbmath.org/1472.140502021-11-25T18:46:10.358925Z"Bertolini, Marina"https://zbmath.org/authors/?q=ai:bertolini.marina"Besana, Gian Mario"https://zbmath.org/authors/?q=ai:besana.gian-mario"Notari, Roberto"https://zbmath.org/authors/?q=ai:notari.roberto"Turrini, Cristina"https://zbmath.org/authors/?q=ai:turrini.cristinaThe main goal of this work is to conclude the analysis of critical loci started in [\textit{M. Bertolini} et al., J. Symb. Comput. 91, 74--97 (2019; Zbl 1403.14070)]. In [loc. cit.] it is shown that the minimal generators of the ideal of the critical locus for 3 projections from \(\mathbb{P}^4\) to \(\mathbb{P}^2\) are cubic polynomials assuming that such minors do not share any common factors. However, in this work the author study the case of 3 projections from \(\mathbb{P}^4\) to \(\mathbb{P}^2\), while dropping the genericity assumptions.
First, the classification of canonical forms of \((n+1)\times n\) matrices, for \(n\leq 3\), of linear forms that drop rank in codimension 1 is introduced. Dropping rank in codimension 1 means that the maximal minors have a non trivial common factor of degree either 1 or 2. Thus, Theorem 2.1 provides all canonical forms of the \(4\times 3\) matrices of linear forms whose maximal minors have a greatest common divisor of degree 1, and Theorem 2.2 when maximal minors have a greatest common divisor of degree 2. Once the classification is done, the authors study the loci where these canonical forms drop rank under some mild generality assumptions and in the main dimensional context of interest for computer vision goals. Thus we arrive at the main theorem of the article, where it is classified the critical locus in the case of three projections from \(\mathbb{P}^4\) to \(\mathbb{P}^2\) in the degenerate case. Finally, last sections introduce the application of their results to the problem of reconstruction in computer vision.Conjecture \(\mathcal{O}\) holds for some horospherical varieties of Picard rank 1https://zbmath.org/1472.140602021-11-25T18:46:10.358925Z"Bones, Lela"https://zbmath.org/authors/?q=ai:bones.lela"Fowler, Garrett"https://zbmath.org/authors/?q=ai:fowler.garrett"Schneider, Lisa"https://zbmath.org/authors/?q=ai:schneider.lisa"Shifler, Ryan M."https://zbmath.org/authors/?q=ai:shifler.ryan-mSummary: Property \(\mathcal{O}\) for an arbitrary complex, Fano manifold \(X\) is a statement about the eigenvalues of the linear operator obtained from the quantum multiplication of the anticanonical class of \(X\). Conjecture \(\mathcal{O}\) is a conjecture that property \(\mathcal{O}\) holds for any Fano variety. Pasquier classified the smooth nonhomogeneous horospherical varieties of Picard rank 1 into five classes. Conjecture \(\mathcal{O}\) has already been shown to hold for the odd symplectic Grassmannians, which is one of these classes. We will show that conjecture \(\mathcal{O}\) holds for two more classes and an example in a third class of Pasquier's list. Perron-Frobenius theory reduces our proofs to be graph-theoretic in nature.The real polynomial eigenvalue problem is well conditioned on the averagehttps://zbmath.org/1472.140682021-11-25T18:46:10.358925Z"Beltrán, Carlos"https://zbmath.org/authors/?q=ai:beltran.carlos"Kozhasov, Khazhgali"https://zbmath.org/authors/?q=ai:kozhasov.khazhgaliThe paper deals with polynomial eigenvalue problem, namely its condition number is studied.
First, the solution variety \(\mathcal{S}\) is introduced, which turns out to be real algebraic or semialgebraic subset of \(\mathbb{R}^m\setminus\{0\}\times S^1\), the product of the variety of inputs and the variety of outputs endowed with Finsler structures.
The condition number \(\mu(a)\) of a given input \(a\) is the sum of the local condition numbers \(\mu(a,x)\) for all solutions for the input.
If the solution variety \(\mathcal{S}\) is co-called nondegenerate, the formula for the squared condition number is proven. Afterwards the formula is used for the case of polynomial eigenvalue problem and so a new proof of the latter is obtained.On totally nonpositive matrices associated with a triple negatively realizablehttps://zbmath.org/1472.150012021-11-25T18:46:10.358925Z"Cantó, Begoña"https://zbmath.org/authors/?q=ai:canto.begona"Cantó, Rafael"https://zbmath.org/authors/?q=ai:canto.rafael"Urbano, Ana María"https://zbmath.org/authors/?q=ai:urbano.ana-mariaSummary: Let \(A \in\mathbb{R}^{n \times n}\) be a totally nonpositive matrix (t.n.p.) with rank \(r\) and principal rank \(p\), that is, every minor of \(A\) is nonpositive and \(p\) is the size of the largest invertible principal submatrix of \(A\). We introduce that a triple \((n, r, p)\) will be called negatively realizable if there exists a t.n.p. matrix \(A\) of order \(n\) and such that its rank is \(r\) and its principal rank is \(p\). In this work we extend the results obtained for irreducible totally nonnegative matrices given in [\textit{R. Cantó} and \textit{A. M. Urbano}, Linear Algebra Appl. 551, 125--146 (2018; Zbl 1415.15001)] to t.n.p. matrices. For that, we consider the sequence of the first \(p\)-indices of \(A\) and study the linear dependence relations between their rows and columns. These relations allow us to construct t.n.p. matrices associated with a triple \((n, r, p)\) negatively realizable and a specific sequence of the first \(p\)-indices.Solving fuzzy dual complex linear systemshttps://zbmath.org/1472.150022021-11-25T18:46:10.358925Z"Chehlabi, M."https://zbmath.org/authors/?q=ai:chehlabi.mSummary: The purpose of this paper is to provide a simple and practical method for finding the solution of the fuzzy complex square systems of order \(n\), included of linear equations which are given in the dual form. For this end, the process of solving a fuzzy dual complex linear system is first described and the conditions of existence and uniqueness of solution is found. Next, the proposed method is appeared with the proof of two theorems and the process of the method is regulated and summarized by solving four real linear square systems of order \(n\). Also, it is shown that the proposed method is efficient and effective in the point of view computationally. Finally, two numerical examples are presented to illustrate the applicability of the method.Generalized core-nilpotent decompositionhttps://zbmath.org/1472.150032021-11-25T18:46:10.358925Z"Karantha, Manjunatha Prasad"https://zbmath.org/authors/?q=ai:karantha.manjunatha-prasad"Varkady, Savitha"https://zbmath.org/authors/?q=ai:varkady.savithaThe authors introduce a number of generalizations of the core-nilpotent decomposition of a square matrix and study their properties. Well-known decompositions, such as the core-EP decomposition and the EP-nilpotent decomposition, are obtained as special cases.Explicit inverse of near Toeplitz pentadiagonal matrices related to higher order difference operatorshttps://zbmath.org/1472.150042021-11-25T18:46:10.358925Z"Kurmanbek, Bakytzhan"https://zbmath.org/authors/?q=ai:kurmanbek.bakytzhan"Erlangga, Yogi"https://zbmath.org/authors/?q=ai:erlangga.yogi-a"Amanbek, Yerlan"https://zbmath.org/authors/?q=ai:amanbek.yerlanSummary: This paper analyzes the inverse of near Toeplitz pentadiagonal matrices, arising from a finite-difference approximation to the fourth-order nonlinear beam equation. Explicit non-recursive inverse matrix formulas and bounds of norms of the inverse matrix are derived for the clamped-free and clamped-clamped boundary conditions. The bound of norms is then used to construct a convergence bound for the fixed-point iteration of the form \(\boldsymbol{u}=f(\boldsymbol{u})\) for solving the nonlinear equation. Numerical computations presented in this paper confirm the theoretical results.The determinant inner product and the Heisenberg product of \(\mathrm{Sym}(2)\)https://zbmath.org/1472.150052021-11-25T18:46:10.358925Z"Crasmareanu, Mircea"https://zbmath.org/authors/?q=ai:crasmareanu.mirceaLet \(A\) be a finite subset of a field and denote by \(D^{n(A)}\) the set of all possible determinants of matrices with entries in \(A\). In this paper, the following problem, typical in additive combinatorics, is investigated: how big is the image set of the determinant function compared to the set \(A\)? Interesting results are obtained, that remain also true also for the set of permanents.Exact determinants of some special circulant matrices involving four kinds of famous numbershttps://zbmath.org/1472.150062021-11-25T18:46:10.358925Z"Jiang, Xiaoyu"https://zbmath.org/authors/?q=ai:jiang.xiaoyu"Hong, Kicheon"https://zbmath.org/authors/?q=ai:hong.kicheonSummary: Circulant matrix family is used for modeling many problems arising in solving various differential equations. The RSFPLR circulant matrices and RSLPFL circulant matrices are two special circulant matrices. The techniques used herein are based on the inverse factorization of polynomial. The exact determinants of these matrices involving Perrin, Padovan, Tribonacci, and the generalized Lucas number are given, respectively.Circulant type matrices with the sum and product of Fibonacci and Lucas numbershttps://zbmath.org/1472.150072021-11-25T18:46:10.358925Z"Jiang, Zhaolin"https://zbmath.org/authors/?q=ai:jiang.zhaolin"Gong, Yanpeng"https://zbmath.org/authors/?q=ai:gong.yanpeng"Gao, Yun"https://zbmath.org/authors/?q=ai:gao.yunSummary: Circulant type matrices have become an important tool in solving differential equations. In this paper, we consider circulant type matrices, including the circulant and left circulant and \(g\)-circulant matrices with the sum and product of Fibonacci and Lucas numbers. Firstly, we discuss the invertibility of the circulant matrix and present the determinant and the inverse matrix by constructing the transformation matrices. Furthermore, the invertibility of the left circulant and \(g\)-circulant matrices is also discussed. We obtain the determinants and the inverse matrices of the left circulant and \(g\)-circulant matrices by utilizing the relation between left circulant, and \(g\)-circulant matrices and circulant matrix, respectively.On minimal bases and indices of rational matrices and their linearizationshttps://zbmath.org/1472.150082021-11-25T18:46:10.358925Z"Amparan, A."https://zbmath.org/authors/?q=ai:amparan.a"Dopico, F. M."https://zbmath.org/authors/?q=ai:dopico.froilan-m"Marcaida, S."https://zbmath.org/authors/?q=ai:marcaida.silvia"Zaballa, I."https://zbmath.org/authors/?q=ai:zaballa.ionSummary: A complete theory of the relationship between the minimal bases and indices of rational matrices and those of their strong linearizations is presented. Such theory is based on establishing first the relationships between the minimal bases and indices of rational matrices and those of their polynomial system matrices under the classical minimality condition and certain additional conditions of properness. This is related to pioneering results obtained by \textit{G. Verghese} [Int. J. Control 31, 1007--1009 (1980; Zbl 0443.93016)], \textit{G. Verghese} et al. [Int. J. Control 30, 235--243 (1979; Zbl 0418.93016)] in 1979--1980, which were the first proving results of this type. It is shown that the definitions of linearizations and strong linearizations do not guarantee any relationship between the minimal bases and indices of the linearizations and the rational matrices in general. In contrast, simple relationships are obtained for the family of strong block minimal bases linearizations, which can be used to compute minimal bases and indices of any rational matrix, including rectangular ones, via algorithms for pencils. These results extend the corresponding ones for other families of linearizations available in recent literature for square rational matrices.``Controlled'' versions of the Collatz-Wielandt and Donsker-Varadhan formulaehttps://zbmath.org/1472.150092021-11-25T18:46:10.358925Z"Arapostathis, Aristotle"https://zbmath.org/authors/?q=ai:arapostathis.aristotle"Borkar, Vivek S."https://zbmath.org/authors/?q=ai:borkar.vivek-shripadSummary: This is an overview of the work of the authors and their collaborators on the characterization of risk-sensitive costs and rewards in terms of an abstract Collatz-Wielandt formula and in case of rewards, also a controlled version of the Donsker-Varadhan formula. For the finite state and action case, this leads to useful linear and dynamic programming formulations for the reward maximization problem in the reducible case.
For the entire collection see [Zbl 1468.60003].The joint spectrumhttps://zbmath.org/1472.150102021-11-25T18:46:10.358925Z"Breuillard, Emmanuel"https://zbmath.org/authors/?q=ai:breuillard.emmanuel"Sert, Cagri"https://zbmath.org/authors/?q=ai:sert.cagriIn this long paper, the authors first introduce the notion of joint spectrum of a compact set of matrices \({\mathcal S}\subset \text{GL}_d({\mathbb C})\), which appears in the context of random matrix products. It is a multi-dimensional generalization of the joint spectral radius introduced by \textit{G.-C. Rota} and \textit{W. G. Strang} [Nederl. Akad. Wet., Proc., Ser. A 63, 379--381 (1960; Zbl 0095.09701)]: \[ R(S) = \lim_{n\to \infty} \sup_{g\in S^n} \|g\|^{1/n}, \] where \(S^n:=\{g_1\cdots g_n: g_i\in S,\ i=1, \dots, n\}\) is the \(n\)-fold product of \(S^n\). The limit \(R(S)\) exists by the submultiplicativity and is independent of the choice of norm \(\|\cdot\|\). When \(S=\{g\}\) is a singleton set, one has the Beruling-Gelfand spectral radius formula which asserts that for each \(g\in \text{GL}_d({\mathbb C})\), \[ \lim_{m\to \infty} \|g^n\|^{1/n} = \rho(g),\] where \(\rho(\cdot )\) is the spectral radius.
Denote by \(a_1(g)\ge \cdots \ge a_d(g)>0\) the singular values of \(g\in \text{GL}_d({\mathbb C})\). The vector \(\kappa(g): = (\log a_1(g), \dots, \log a_d(g))\in {\mathbb R}^d\) is called the Cartan vector of \(g\) and \(\kappa(S) := \{\kappa(g): g\in S\}\) is clearly a compact set. It turns out that if the subgroup \(\Gamma\) that \(S\) generates acts irreducibly on \({\mathbb C}^d\), then \(\frac 1n\kappa (S^n)\) converges in Hausdorff metric to a compact subset \(J(S)\) of \({\mathbb R}^d\) and \(J(S)\) is called the joint spectrum of \(S\). They give a thorough study of \(J(S)\) and find that several classical properties of the joint spectral radius hold in the generalized setting. Several other topics are studied. A discussion on the analogue of the Lagarias-Wang finiteness conjecture is given. Examples are worked out. The paper relies extensively on color figures to help the reader.Bounds on the spectrum of nonsingular triangular \((0,1)\)-matriceshttps://zbmath.org/1472.150112021-11-25T18:46:10.358925Z"Kaarnioja, Vesa"https://zbmath.org/authors/?q=ai:kaarnioja.vesaLet \(K_n\) denote the set of all invertible \(n\times n\) lower triangular \((0,1)\)-matrices, and let \(\lambda_n(\cdot)\) and \(\lambda_1(\cdot)\) be the smallest and, respectively, largest eigenvalue of a given \(n\times n\) matrix with real eigenvalues. The numbers
\[
c_n=\min\,\{\lambda_n(XX^T): X\in K_n\}\quad\mathrm{and}\quad C_n=\max\,\{\lambda_1(XX^T): X\in K_n\}
\]
are useful in studying extremal eigenvalues of certain GCD (greatest common divisor) and LCM (least common multiple) matrices and their generalizations. \textit{S. Hong} and \textit{R. Loewy} [Glasgow Math. J. 46, 551--569 (2004; Zbl 1083.11021)] introcuced \(c_n\). \textit{P. Ilmonen} et al. [Linear Algebra Appl. 429, 859--874 (2008; Zbl 1143.15016)] introduced \(C_n\).
Furthermore, \textit{M. Mattila} [Linear Algebra Appl. 466, 1--20 (2015; Zbl 1395.15028)] found a lower bound for \(c_n\), and the reviewer (Appendix of [\textit{E. Altınışık} et al., Linear Algebra Appl. 493, 1--13 (2016; Zbl 1334.15079)]) improved it.
The present author gives the following bound and demonstrates by numerical experiments that it improves significantly the aforementioned bounds. If \(n\) is odd, then
\[
c_n\ge\big(\tfrac{1}{25}\phi^{-4n}+\tfrac{2}{25}\phi^{-2n}-\tfrac{2}{5\sqrt{5}}n\phi^{-2n} -\tfrac{23}{25}+n+\tfrac{2}{25}\phi^{2n}+\tfrac{2}{5\sqrt{5}}n\phi^{2n}+ \tfrac{1}{25}\phi^{4n}\big)^{-\frac{1}{2}},
\]
where \(\phi\) denotes the golden ratio. If \(n\) is even, then the coefficient of the second and sixth term is \(\frac{4}{25}\), and the fourth term is \(-\frac{2}{5}\). He also conjectures that asymptotically \(c_n\sim 5\phi^{-2n}\). Note that later on \textit{R. Loewy} [Linear Algebra Appl. 608, 203--213 (2021; Zbl 07309772)] proved this.
Ilmonen et al. [loc. cit.] found an upper bound for \(C_n\). The present author proves the explicit formula
\[
C_n=\frac{1}{4}\csc^2\left(\frac{\pi}{4n+2}\right)=\frac{(2n+1)^2}{\pi^2}+\frac{1}{12}+O\big(\frac{1}{n^2}\big).
\]Global properties of eigenvalues of parametric rank one perturbations for unstructured and structured matriceshttps://zbmath.org/1472.150122021-11-25T18:46:10.358925Z"Ran, André C. M."https://zbmath.org/authors/?q=ai:ran.andre-c-m"Wojtylak, Michał"https://zbmath.org/authors/?q=ai:wojtylak.michalThe paper contributes to eigenvalue perturbation theory. The authors previously considered perturbations of the form \(A+tuv^*\), where \(t\in \mathbb R\), in [\textit{A. C. M. Ran} and \textit{M. Wojtylak}, Linear Algebra Appl. 437, No. 2, 589--600 (2012; Zbl 1247.15009)]. Here, they also consider angular perturbations \(A+e^{i\theta}uv^*\) where \(\theta\in [0,2\pi)\). Connecting these two cases, the authors prove new results on the global behaviour of the eigenvalues. The two main problems under consideration are defining the eigenvalues as functions of the parameter \(\tau\) (where either \(\tau\in \mathbb R\) or \(\tau=e^{i\theta}\)), in which case the eigenvalues can be defined as analytic functions, and considering the case of large \(|\tau|\rightarrow \infty\), where the eigenvalues are studied through the roots of the polynomial \(m_A(\lambda)-\tau p_{uv}(\lambda)\), where \(m_A(\lambda)\) is the minimal polynomial of \(A\) and \(p_{uv}(\lambda)=v^*m_A(\lambda)(\lambda I_n-A)^{-1}u\). The results are applied to various families of matrices.A new random perturbation interval of symmetric eigenvalue problemhttps://zbmath.org/1472.150132021-11-25T18:46:10.358925Z"Tang, Ling"https://zbmath.org/authors/?q=ai:tang.ling"Xiao, Chuanfu"https://zbmath.org/authors/?q=ai:xiao.chuanfu"Li, Hanyu"https://zbmath.org/authors/?q=ai:li.hanyuSummary: A large interval of random perturbation is presented in this study. Under the derived interval, the probability that the simple eigenvalue of original symmetric matrix is still simple is not less than a given confidence level. Numerical examples are provided to illustrate the obtained results.Rational forms for block lower-triangular matriceshttps://zbmath.org/1472.150142021-11-25T18:46:10.358925Z"Dewilde, Patrick"https://zbmath.org/authors/?q=ai:dewilde.patrick-mThe rational forms representing linear time invariant (LTI) systems exhibit important properties of the systems themselves. Here, the author seeks similar forms for linear time-variant (LTV) systems, which are expected to have comparable properties, and tries to generalize the LTI case. It turns out such forms exist and the theory given in this paper yields representations for block lower-triangular matrices as ratios of (block) echelon matrices. A Euclidean algorithm is lacking and the presented theory makes use of a technique derived from [\textit{P. van Dooren}, BIT 24, 681--699 (1984; Zbl 0549.93019)]. Applications in several areas are found. At the end, a conjecture is presented.Products of positive definite symplectic matriceshttps://zbmath.org/1472.150152021-11-25T18:46:10.358925Z"Granario, Daryl Q."https://zbmath.org/authors/?q=ai:granario.daryl-q"Tam, Tin-Yau"https://zbmath.org/authors/?q=ai:tam.tin-yauLet us recall that the complex symplectic group \(\mathrm{Sp}(2n,\mathbb C)\) is defined as follows: \[\mathrm{Sp}(2n,\mathbb C)=\left\{A\in\mathrm{GL}(2n,\mathbb C): A^TJ_nA=J_n\right\}\text{ with }J_n=\begin{bmatrix}0 & I_n\\
-I_n & 0\\
\end{bmatrix}, \] and that a matrix is called positive definite if its spectrum is contained in \((0,\infty)\).
\textit{C. S. Ballantine} [Pac. J. Math. 23, 427--433 (1968; Zbl 0211.35302); J. Algebra 10, 174--182 (1968; Zbl 0225.15012); Linear Algebra Appl. 3, 79--114 (1970; Zbl 0192.37002)] proved that every matrix from \(\mathrm{GL}(n,\mathbb C)\) with a positive determinant is the product of five positive definite matrices.
In the present work, the authors study the same problem in \(\mathrm{Sp}(2n,\mathbb C)\). They show that every symplectic matrix is the product of five symplectic positive definite matrices. The proof uses canonical forms of symplectic matrices. It is also shown that there exist symplectic matrices that cannot be written as a product of a lower number of symplectic positive definite matrices. Additionally, the authors characterize products of two symplectic positive definite matrices.
The paper contains a very comprehensive introduction into the topic and an up-to-date list of references.An alternative canonical form for quaternionic \(H\)-unitary matriceshttps://zbmath.org/1472.150162021-11-25T18:46:10.358925Z"Groenewald, G. J."https://zbmath.org/authors/?q=ai:groenewald.gilbert-j"Janse van Rensburg, D. B."https://zbmath.org/authors/?q=ai:janse-van-rensburg.dawie-b"Ran, A. C. M."https://zbmath.org/authors/?q=ai:ran.andre-c-mLet $H$ be the quaternion ring. Let $(A,H)$ be a pair of matrices over $H$. The matrix $A$ is \(H\)-unitary if $H=H^*$ is invertible and $A^*HA=H$. The authors find an invertible matrix $S$ such that the transformations from $(A,H)$ to $(S^{-1}AS,S^*HS)$ brings the matrix A in Jordan form and simultaneously brings $H$ into a canonical form. The authors found inspiration for their work in the results in [the authors, Oper. Matrices 10, No. 4, 739--783 (2016; Zbl 1360.15015); Oper. Theory: Adv. Appl. 271, 269--290 (2018; Zbl 07137673)]. Their main goal is the study of a quaternionic pair of matrices, a topic that is still under development (see [\textit{L. Rodman}, Topics in quaternion linear algebra. Princeton, NJ: Princeton University Press (2014; Zbl 1304.15004)]).Palindromic linearizations of palindromic matrix polynomials of odd degree obtained from Fiedler-like pencilshttps://zbmath.org/1472.150172021-11-25T18:46:10.358925Z"Das, Ranjan Kumar"https://zbmath.org/authors/?q=ai:das.ranjan-kumar"Alam, Rafikul"https://zbmath.org/authors/?q=ai:alam.rafikulSummary: Palindromic matrix polynomials arise in many applications. Structure-preserving linearizations of palindromic matrix polynomials have been proposed in the literature so as to preserve the spectral symmetry in the eigenvalues. We present a new family of palindromic strong linearizations of a palindromic matrix polynomial of odd degree. A salient feature of the new family is that it allows the construction of banded palindromic linearizations of block-bandwidth \(k+1\) for any \(k=0:m-2\), where \(m\) is the degree of the palindromic matrix polynomial. Low bandwidth palindromic pencils may be useful for numerical computations. Our construction of the new family is based on Fiedler companion matrices associated with matrix polynomials and the construction is operation-free. Moreover, the new family of palindromic pencils allows operation-free recovery of eigenvectors and minimal bases, and an easy recovery of minimal indices of matrix polynomials from those of the palindromic linearizations. We also present an operation-free algorithm for construction of palindromic pencils belonging to the new family.Rank one perturbations of matrix pencilshttps://zbmath.org/1472.150182021-11-25T18:46:10.358925Z"Dodig, Marija"https://zbmath.org/authors/?q=ai:dodig.marija"Stošić, Marko"https://zbmath.org/authors/?q=ai:stosic.markoUsing some of their earlier results (see for example [the first author, Linear Algebra Appl. 438, No. 8, 3155--3173 (2013; Zbl 1269.15029)]), the authors completely resolve the open problem of describing all possible Kronecker invariants of an arbitrary matrix pencil under rank one perturbations. Their solution is explicit and constructive, and is valid for arbitrary pencils. They combine results on one-row matrix pencil completions with combinatorial results on double generalized majorization, and develop some new techniques.Exact conditions for preservation of the partial indices of a perturbed triangular \(2 \times 2\) matrix functionhttps://zbmath.org/1472.150192021-11-25T18:46:10.358925Z"Adukov, Victor M."https://zbmath.org/authors/?q=ai:adukov.viktor-mikhailovich|adukov.viktor-michailovich"Mishuris, Gennady"https://zbmath.org/authors/?q=ai:mishuris.gennady-s"Rogosin, Sergei V."https://zbmath.org/authors/?q=ai:rogosin.sergei-vSummary: The possible instability of partial indices is one of the important constraints in the creation of approximate methods for the factorization of matrix functions. This paper is devoted to a study of a specific class of triangular matrix functions given on the unit circle with a stable and unstable set of partial indices. Exact conditions are derived that guarantee a preservation of the unstable set of partial indices during a perturbation of a matrix within the class. Thus, even in this probably simplest of cases, when the factorization technique is well developed, the structure of the parametric space (guiding the types of matrix perturbations) is non-trivial.On explicit Wiener-Hopf factorization of \(2 \times 2\) matrices in a vicinity of a given matrixhttps://zbmath.org/1472.150202021-11-25T18:46:10.358925Z"Ephremidze, L."https://zbmath.org/authors/?q=ai:ephremidze.lasha"Spitkovsky, I."https://zbmath.org/authors/?q=ai:spitkovsky.ilya-matveySummary: As it is known, the existence of the Wiener-Hopf factorization for a given matrix is a well-studied problem. Severe difficulties arise, however, when one needs to compute the factors approximately and obtain the partial indices. This problem is very important in various engineering applications and, therefore, remains to be subject of intensive investigations. In the present paper, we approximate a given matrix function and then explicitly factorize the approximation regardless of whether it has stable partial indices. For this reason, a technique developed in the Janashia-Lagvilava matrix spectral factorization method is applied. Numerical simulations illustrate our ideas in simple situations that demonstrate the potential of the method.On recursive computation of coprime factorizations of rational matriceshttps://zbmath.org/1472.150212021-11-25T18:46:10.358925Z"Varga, Andreas"https://zbmath.org/authors/?q=ai:varga.andreasSummary: General computational methods based on descriptor state-space realizations are proposed to compute coprime factorizations of rational matrices with minimum degree denominators. The new methods rely on recursive pole dislocation techniques, which allow to successively place all poles of the factors into a ``good'' region of the complex plane. The resulting McMillan degree of the denominator factor is equal to the number of poles lying in the complementary ``bad'' region and therefore is minimal. The developed pole dislocation techniques are instrumental for devising numerically reliable procedures for the computation of coprime factorizations with proper and stable factors of arbitrary improper rational matrices and coprime factorizations with inner denominators. Implementation aspects of the proposed algorithms are discussed and illustrative examples are given.The generalized bisymmetric (bi-skew-symmetric) solutions of a class of matrix equations and its least squares problemhttps://zbmath.org/1472.150222021-11-25T18:46:10.358925Z"Ke, Yifen"https://zbmath.org/authors/?q=ai:ke.yifen"Ma, Changfeng"https://zbmath.org/authors/?q=ai:ma.changfengSummary: The solvability conditions and the general expression of the generalized bisymmetric and bi-skew-symmetric solutions of a class of matrix equations \((A X = B\), \(X C = D)\) are established, respectively. If the solvability conditions are not satisfied, the generalized bisymmetric and bi-skew-symmetric least squares solutions of the matrix equations are considered. In addition, two algorithms are provided to compute the generalized bisymmetric and bi-skew-symmetric least squares solutions. Numerical experiments illustrate that the results are reasonable.New sufficient conditions for the unique solution of a square Sylvester-like absolute value equationhttps://zbmath.org/1472.150232021-11-25T18:46:10.358925Z"Wang, Li-Min"https://zbmath.org/authors/?q=ai:wang.limin|wang.limin.1"Li, Cui-Xia"https://zbmath.org/authors/?q=ai:li.cuixiaAssume that \(A, C\in\mathbb{R}^{m\times m}\), \(B, D\in\mathbb{R}^{n\times n}\) and \(F\in\mathbb{R}^{m\times n}\) are matrices and \(X\in\mathbb{R}^{m\times n}\) is an unknown matrix. The Sylvester-like absolute valued equation reads \[ AXB+C|X|D=F\,. \] Here, the authors give two sufficient conditions for the existence of a unique solution of the above equation. Some special cases are also considered.Corrigendum to: ``The Gerstenhaber problem for commuting triples of matrices is `decidable' ''https://zbmath.org/1472.150242021-11-25T18:46:10.358925Z"O'Meara, Kevin C."https://zbmath.org/authors/?q=ai:omeara.kevin-cSummary: I correct a slip in an argument in the above paper [the author, ibid. 48, No. 2, 453--466 (2020; Zbl 1436.15018)]. This does not affect the
main result: the Gerstenhaber Problem is Turing computable for all fields.A note on completion to the unitary matriceshttps://zbmath.org/1472.150252021-11-25T18:46:10.358925Z"de Andrade Bezerra, Johanns"https://zbmath.org/authors/?q=ai:de-andrade-bezerra.johannsSummary: Let \(U=( \begin{smallmatrix} {U_1}&{U_2}\\ {U_3}&{U_4} \end{smallmatrix})\) be a complex matrix of order \(n\). In this paper, once a square submatrix \(U_1\) is fixed, we present several necessary conditions on \(U_1, U_2, U_3\) and \(U_4\), where \(U\) is a unitary matrix. Particularly, given a unitary matrix \(U=( \begin{smallmatrix} {U_1}&{U_2}\\ {U_3}&{U_4} \end{smallmatrix})\), we characterize the completions \((\begin{smallmatrix} {U_1}&X\\ Y&{U_4} \end{smallmatrix})\) and \((\begin{smallmatrix} {U_1}&X\\ Y&Z \end{smallmatrix})\) to some unitary matrix, whenever \(U_1\) and \(U_4\) are partial isometries.Single-exponential bounds for the smallest singular value of Vandermonde matrices in the sub-Rayleigh regimehttps://zbmath.org/1472.150262021-11-25T18:46:10.358925Z"Batenkov, Dmitry"https://zbmath.org/authors/?q=ai:batenkov.dmitry"Goldman, Gil"https://zbmath.org/authors/?q=ai:goldman.gilSummary: Following recent interest by the community, the scaling of the minimal singular value of a Vandermonde matrix with nodes forming clusters on the length scale of Rayleigh distance on the complex unit circle is studied. Using approximation theoretic properties of exponential sums, we show that the decay is only single exponential in the size of the largest cluster, and the bound holds for arbitrary small minimal separation distance. We also obtain a generalization of well-known bounds on the smallest eigenvalue of the generalized prolate matrix in the multi-cluster geometry. Finally, the results are extended to the entire spectrum.On some inequalities for accretive-dissipative matriceshttps://zbmath.org/1472.150272021-11-25T18:46:10.358925Z"Mao, Yanling"https://zbmath.org/authors/?q=ai:mao.yanling"Liu, Xilan"https://zbmath.org/authors/?q=ai:liu.xilan\textit{F. Kittaneh} and \textit{M. Sakkijha} [Linear Multilinear Algebra 67, No. 5, 1037--1042 (2019; Zbl 1411.15012)] established a double inequality for any accretive-dissipative block matrix comparing the norm of its diagonal entries and the norm of its off-diagonal entries. Here, the authors employ a matrix decomposition for sector matrices due to \textit{F. Zhang} [Linear Multilinear Algebra 63, No. 10, 2033--2042 (2015; Zbl 1361.15030)] to extend the aforementioned result.On effective criterion of stability of partial indices for matrix polynomialshttps://zbmath.org/1472.150282021-11-25T18:46:10.358925Z"Adukova, N. V."https://zbmath.org/authors/?q=ai:adukova.natalya-viktorovna"Adukov, V. M."https://zbmath.org/authors/?q=ai:adukov.viktor-mikhailovich|adukov.viktor-michailovichSummary: In the work, we obtain an effective criterion of the stability of the partial indices for matrix polynomials under an arbitrary sufficiently small perturbation. Verification of the stability is reduced to calculation of the ranks for two explicitly defined Toeplitz matrices. Furthermore, we define a notion of the stability of the partial indices in the given class of matrix functions. This means that we will consider an allowable small perturbation such that a perturbed matrix function belong to the same class as the original one. We prove that in the class of matrix polynomials the Gohberg-Krein-Bojarsky criterion is preserved, i.e. new stability cases do not arise. Our proof of the stability criterion in this class does not use the Gohberg-Krein-Bojarsky theorem.On the geometry of numerical ranges over finite fieldshttps://zbmath.org/1472.150292021-11-25T18:46:10.358925Z"Camenga, Kristin A."https://zbmath.org/authors/?q=ai:camenga.kristin-a"Collins, Brandon"https://zbmath.org/authors/?q=ai:collins.brandon"Hoefer, Gage"https://zbmath.org/authors/?q=ai:hoefer.gage"Quezada, Jonny"https://zbmath.org/authors/?q=ai:quezada.jonny"Rault, Patrick X."https://zbmath.org/authors/?q=ai:rault.patrick-x"Willson, James"https://zbmath.org/authors/?q=ai:willson.james-k"Yates, Rebekah B. Johnson"https://zbmath.org/authors/?q=ai:yates.rebekah-b-johnsonSummary: Numerical ranges over a certain family of finite fields were classified in 2016 by a team including our fifth author [\textit{J. I. Coons} et al., Linear Algebra Appl. 501, 37--47 (2016; Zbl 1334.15054)]. Soon afterward \textit{E. Ballico} generalized these results to all finite fields and published some new results about the cardinality of the finite field numerical range [Linear Algebra Appl. 512, 162--171 (2017; Zbl 1353.15028); Linear Algebra Appl. 556, 421--427 (2018; Zbl 1442.15051)]. In this paper we study the geometry of these finite fields using the boundary generating curve, first introduced by \textit{R. Kippenhahn} in 1951 [Math. Nachr. 6, 193--228 (1951; Zbl 0044.16201); Linear Multilinear Algebra 56, No. 1--2, 185--225 (2008; Zbl 1137.47003); translation from Math. Nachr. 6, 193--228 (1951)]. We restrict our study to square matrices of dimension 2, with at least one eigenvalue in \(\mathbb{F}_{q^2}\).Nearest \(\Omega \)-stable matrix via Riemannian optimizationhttps://zbmath.org/1472.150302021-11-25T18:46:10.358925Z"Noferini, Vanni"https://zbmath.org/authors/?q=ai:noferini.vanni"Poloni, Federico"https://zbmath.org/authors/?q=ai:poloni.federico-gSummary: We study the problem of finding the nearest \(\varOmega \)-stable matrix to a certain matrix \(A\), i.e., the nearest matrix with all its eigenvalues in a prescribed closed set \(\varOmega \). Distances are measured in the Frobenius norm. An important special case is finding the nearest Hurwitz or Schur stable matrix, which has applications in systems theory. We describe a reformulation of the task as an optimization problem on the Riemannian manifold of orthogonal (or unitary) matrices. The problem can then be solved using standard methods from the theory of Riemannian optimization. The resulting algorithm is remarkably fast on small-scale and medium-scale matrices, and returns directly a Schur factorization of the minimizer, sidestepping the numerical difficulties associated with eigenvalues with high multiplicity.On constant-trace representations of degenerate Clifford algebrashttps://zbmath.org/1472.150312021-11-25T18:46:10.358925Z"Mahmoudi, M. G."https://zbmath.org/authors/?q=ai:mahmoudi.mohammad-gholamzadeh"Sidhwa, H. H."https://zbmath.org/authors/?q=ai:sidhwa.h-hSummary: In this paper, we complement some recent results of \textit{L. Márki} et al. [Isr. J. Math. 208, 373--384 (2015; Zbl 1344.16023)], by investigating the constant-trace representations of a Clifford algebra \(C(V)\) of an arbitrary quadratic form \(q:V\rightarrow F\) (possibly degenerate) and we present some relevant applications. In particular, the existence of the polynomial identities of \(C(V)\) of particular form when the characteristic of the base field is zero is looked at. Furthermore, a lower bound is found on the minimal number \(t\), such that \(C(V)\) can be embedded in a matrix ring of degree \(t\), over some commutative \(F\)-algebra. Also, some results on the dimension of commutative subalgebras of \(C(V)\) are obtained.Generalizing classical Clifford algebras, graded Clifford algebras and their associated geometryhttps://zbmath.org/1472.150322021-11-25T18:46:10.358925Z"Vancliff, Michaela"https://zbmath.org/authors/?q=ai:vancliff.michaelaSummary: This article is based on a talk given by the author at the \textit{12th International Conference on Clifford Algebras and their Applications in Mathematical Physics}. A generalization, introduced by \textit{T. Cassidy} and the author [J. Lond. Math. Soc., II. Ser. 90, No. 2, 631--636 (2014; Zbl 1303.16029)]
of a classical Clifford algebra is discussed together with connections between that generalization and a generalization of a graded Clifford algebra. A geometric approach to studying the algebras, viewed through the lens of Artin, Tate and Van den Bergh's noncommutative algebraic geometry, is also presented.On Jordan-Clifford algebras, three fermion generations with Higgs fields and a \(\mathrm{SU}(3) \times \mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{U}(1)\) modelhttps://zbmath.org/1472.150332021-11-25T18:46:10.358925Z"Castro Perelman, Carlos"https://zbmath.org/authors/?q=ai:castro-perelman.carlosSummary: Previously we have shown that the algebra
\[
J_3 [{\mathbb{C}}\otimes{\mathbb{O}}] \otimes C \ell (4,{\mathbb{C}}),
\]
given by the tensor product of the complex exceptional Jordan \(J_3 [\mathbb{C}\otimes\mathbb{O}]\) and the complex Clifford algebra \(C \ell (4,\mathbb{C})\), can describe all of the spinorial degrees of freedom of three generations of fermions in four-space-time dimensions. We extend our construction to show that it also includes the degrees of freedom of three sets of pairs of complex scalar Higgs-doublets \(\{\mathbf{H}^{(m)}_L, \mathbf{H}^{(m)}_R\}\); \(m = 1,2,3\), and their \(\mathrm{CPT}\) conjugates. Furthermore, a close inspection of the fermion structure of each generation reveals that it fits naturally with the sixteen complex-dimensional representation of the internal left/right symmetric gauge group \(G_{LR} = \mathrm{SU}(3)_C \times \mathrm{SU}(2)_L \times \mathrm{SU}(2)_R \times \mathrm{U}(1)\). It is reviewed how the latter group emerges from the intersection of \(\mathrm{SO}(10)\) and \(\mathrm{SU}(3) \times \mathrm{SU}(3) \times \mathrm{SU}(3)\) in \(E_6\). In the concluding remarks we briefly discuss the role that the extra Higgs fields may have as dark matter candidates; the construction of Chern-Simons-like matrix cubic actions; hexaquarks; supersymmetry and Clifford bundles over the complex-octonionic projective plane \((\mathbb{C}\otimes\mathbb{O}) \mathbb{P}^2\) whose isometry group is \(E_6\).On the problem of choosing subgroups of Clifford algebras for applications in fundamental physicshttps://zbmath.org/1472.150342021-11-25T18:46:10.358925Z"Wilson, Robert Arnott"https://zbmath.org/authors/?q=ai:wilson.robert-aSummary: Clifford algebras are used for constructing spin groups, and are therefore of particular importance in the theory of quantum mechanics. An algebraist's perspective on the many subgroups and subalgebras of Clifford algebras may suggest ways in which they might be applied more widely to describe the fundamental properties of matter. I do not claim to build a physical theory on top of the fundamental algebra, and my suggestions for possible physical interpretations are indicative only, and may not work. Nevertheless, both the existence of three generations of fermions and the symmetry-breaking of the weak interaction seem to emerge naturally from an extension of the Dirac algebra from complex numbers to quaternions.Numerical study on Moore-Penrose inverse of tensors via Einstein producthttps://zbmath.org/1472.150352021-11-25T18:46:10.358925Z"Huang, Baohua"https://zbmath.org/authors/?q=ai:huang.baohuaSummary: The notation of Moore-Penrose inverse of matrices has been extended from matrix space to even-order tensor space with Einstein product. In this paper, we give the numerical study on the Moore-Penrose inverse of tensors via the Einstein product. More precisely, we transform the calculation of Moore-Penrose inverse of tensors via the Einstein product into solving a class of tensor equations via the Einstein product. Then, by means of the conjugate gradient method, we obtain the approximate Moore-Penrose inverse of tensors via the Einstein product. Finally, we report some numerical examples to show the efficiency of the proposed methods and testify the conclusion suggested in this paper.On the rank and the approximation of symmetric tensorshttps://zbmath.org/1472.150362021-11-25T18:46:10.358925Z"Rodríguez, Jorge Tomás"https://zbmath.org/authors/?q=ai:rodriguez.jorge-tomasSummary: In this work we study different notions of ranks and approximation of tensors. We consider the tensor rank, the nuclear rank and we introduce the notion of symmetric decomposable rank, a notion of rank defined only on symmetric tensors. We show that when approximating symmetric tensors, using the symmetric decomposable rank has some significant advantages over the tensor rank and the nuclear rank.Bilinear factorizations subject to monomial equality constraints via tensor decompositionshttps://zbmath.org/1472.150372021-11-25T18:46:10.358925Z"Sørensen, Mikael"https://zbmath.org/authors/?q=ai:sorensen.mikael"De Lathauwer, Lieven"https://zbmath.org/authors/?q=ai:de-lathauwer.lieven"Sidiropoulos, Nicholaos D."https://zbmath.org/authors/?q=ai:sidiropoulos.nicholaos-dThe authors consider a problem of uniqueness of a tensor decomposition, related to tensors \(\mathcal X\) of type \(I\times J\times K\). In particular, they are interested in decompositions which arise by unfolding \(\mathcal X\) to a matrix \(X\) and considering representations of type \(X=(A\odot B)S^t\), where \( A\), \(B\), \(S\) are matrices of size \(I\times R\), \(J\times R\), \(K\times R\), respectively, and \(\odot\) represents the column-wise Kronecker product. The authors consider the uniqueness of the decomposition when the entries of the matrix \(A\) are subject to monomial equality constraints, i.e., the entries must satisfy equations of type \(a_{p_1r}\cdots a_{p_Lr} = a_{s_1r}\cdots a_{s_Lr}\), and similar constraints hold on the entries of \(S\). Such monomial equality constraints arise naturally in decomposition problems related to signal processing or binary latent variable modeling. The authors extend existing methods for detecting the uniqueness of a general rank decomposition to tensor decompositions with monomial equality constraints. The block term decomposition method allows to restrict the monomial constraints to rank constraint. Thus, the authors are able to find several conditions that guarantee the uniqueness of the decomposition. The authors observe that the same methods can provide uniqueness results even for weighted decompositions of type \(X=(D*A\odot B)S^t\), where \(D\) is a matrix of weights, and \(*\) denotes the entry-wise (Hadamard) product.Exact matrix completion based on low rank Hankel structure in the Fourier domainhttps://zbmath.org/1472.150382021-11-25T18:46:10.358925Z"Chen, Jinchi"https://zbmath.org/authors/?q=ai:chen.jinchi"Gao, Weiguo"https://zbmath.org/authors/?q=ai:gao.weiguo"Wei, Ke"https://zbmath.org/authors/?q=ai:wei.keSummary: Matrix completion is about recovering a matrix from its partial revealed entries, and it can often be achieved by exploiting the inherent simplicity or low dimensional structure of the target matrix. For instance, a typical notion of matrix simplicity is low rank. In this paper we study matrix completion based on another low dimensional structure, namely the low rank Hankel structure in the Fourier domain. It is shown that matrices with this structure can be exactly recovered by solving a convex optimization program provided the sampling complexity is nearly optimal. Empirical results are also presented to justify the effectiveness of the convex method.Linear maps preserving the Lorentz-cone spectrum in certain subspaces of \(M_n\)https://zbmath.org/1472.150392021-11-25T18:46:10.358925Z"Bueno, M. I."https://zbmath.org/authors/?q=ai:bueno.maribel-i|bueno.maria-isabel"Furtado, S."https://zbmath.org/authors/?q=ai:furtado.susana"Sivakumar, K. C."https://zbmath.org/authors/?q=ai:sivakumar.koratti-chengalrayanLet \(n \geq 3\) and consider the Lorentz cone \(\mathcal K = \{(x,x_n)\in \mathbb R^{n-1}\times \mathbb R : \lVert x \rVert \leq x_n \}\), where \(\lVert x\rVert\) is the 2-norm of \(x\). It is well-known that the Lorentz cone \(\mathcal K\) is self-dual, and so the eigenvalue complementarity problem associated with \(\mathcal K\) is, for a given real-valued \(n \times n\) matrix \(A\), finding a scalar \(\lambda \in \mathbb R\) and nonzero vector \(x \in \mathbb R^n\) such that \[x \in \mathcal K, \qquad (A-\lambda I_n)x \in \mathcal K, \qquad x^T(A-\lambda I_n)x = 0,\] where \(I_n\) denotes the \(n \times n\) matrix and \(x^T\) denotes the transpose of \(x\). The scalar \(\lambda\) is called \textit{Lorentz eigenvalue} of \(A\) and the set of all Lorentz eigenvalues of \(A\) is denoted \(\sigma_\mathcal{K}(A)\), and called the \textit{Lorentz cone spectrum of \(A\)}. The authors explore linear preserver problems related to the Lorentz cone spectrum of matrices. In other words, they obtain characterizations of linear maps \(\phi: \mathcal M \to \mathcal M\) such that \(\sigma_\mathcal{K}(\phi(A)) = \sigma_\mathcal{K}(A)\) for all \(A \in \mathcal M\), where \(\mathcal M\) is a given subspace of \(M_n(\mathbb R)\).
The authors call maps of the form \(\phi(A) = PAQ\) or \(\phi(A) = PA^TQ\), where \(P\) and \(Q\) are fixed matrices, \textit{standard linear maps} (most linear preserver problems in the literature conclude that the maps have this particular form; see [\textit{C.-K. Li} and \textit{S. Pierce}, Am. Math. Mon. 108, No. 7, 591--605 (2001; Zbl 0991.15001)] for a general description). Lorentz cone spectrum preservers are considered on the following subspaces \(\mathcal M\): diagonal matrices, block-diagonal matrices of the form \(\tilde{A}\oplus[a]\) with \(a \in \mathbb{R}\) and \(\tilde{A}\in M_{n-1}(\mathbb{R})\) symmetric, block-diagonal matrices of the form \(\tilde{A}\oplus[a]\) with \(a \in \mathbb{R}\) and \(\tilde{A}\in M_{n-1}(\mathbb{R})\) arbitrary, the space of symmetric matrices, and the full matrix space \(M_n(\mathbb R)\). For the diagonal and block-diagonal spaces, the authors show (see Theorem 4.2) that Lorentz cone spectrum preservers of these spaces are, in fact, standard maps with \(P\) and \(Q\) invertible of a specified form.
For the symmetric and full matrix space case, the complete description of standard linear maps that preserve the Lorentz cone spectrum are provided. It seems to be an open problem as to whether or not all Lorentz cone spectrum preservers of the symmetric and full matrix spaces must in fact be standard maps.
As a byproduct of studying preservers in this paper, certain properties of the Lorentz cone spectrum (and the interplay with the usual spectrum) are developed for special classes of matrices; in particular, rank-one matrices.Linear preservers of copositive matriceshttps://zbmath.org/1472.150402021-11-25T18:46:10.358925Z"Furtado, Susana"https://zbmath.org/authors/?q=ai:furtado.susana"Johnson, C. R."https://zbmath.org/authors/?q=ai:johnson.charles-royal"Zhang, Yulin"https://zbmath.org/authors/?q=ai:zhang.yulinSummary: An \(n\)-by-\(n\) real symmetric matrix is called copositive if its quadratic form is nonnegative on nonnegative vectors. Our interest is in identifying which linear transformations on symmetric matrices preserve copositivity either in the into or onto sense. We conjecture that in the onto case, the map must be congruence by a monomial matrix (a permutation times a positive diagonal matrix). This is proven under each of some additional natural assumptions. Also, the into preservers of standard type are characterized. A general characterization in the into case seems difficult, and examples are given. One of them provides a counterexample to a conjecture about the into preservers.Optimal rank-1 Hankel approximation of matrices: Frobenius norm and spectral norm and Cadzow's algorithmhttps://zbmath.org/1472.150412021-11-25T18:46:10.358925Z"Knirsch, Hanna"https://zbmath.org/authors/?q=ai:knirsch.hanna"Petz, Markus"https://zbmath.org/authors/?q=ai:petz.markus"Plonka, Gerlind"https://zbmath.org/authors/?q=ai:plonka.gerlindSummary: We characterize optimal rank-1 matrix approximations with Hankel or Toeplitz structure with regard to two different norms, the Frobenius norm and the spectral norm, in a new way. More precisely, we show that these rank-1 matrix approximation problems can be solved by maximizing special rational functions. Our approach enables us to show that the optimal solutions with respect to these two norms have completely different structure and only coincide in the trivial case when the singular value decomposition already provides an optimal rank-1 approximation with the desired Hankel or Toeplitz structure. We also prove that the Cadzow algorithm for structured low-rank approximations always converges to a fixed point in the rank-1 case. However, it usually does not converge to the optimal solution, neither with regard to the Frobenius norm nor the spectral norm.Weighted minimization problems for quaternion matriceshttps://zbmath.org/1472.150422021-11-25T18:46:10.358925Z"Kyrchei, Ivan I."https://zbmath.org/authors/?q=ai:kyrchei.ivan"Mosić, Dijana"https://zbmath.org/authors/?q=ai:mosic.dijana"Stanimirović, Predrag S."https://zbmath.org/authors/?q=ai:stanimirovic.predrag-sSummary: We introduce three kinds of new weighted quaternion-matrix minimization problems in order to extend some well-known constrained approximation problems. The main result is the claim that these new minimization problems have unique solutions which are expressed in terms of expressions involving weighted core-EP inverse and its dual for adequate quaternion matrices. Also, determinantal expressions of solutions for considered minimization problems are given. Obtained theoretical results are justified by a numerical example.Singularity of random symmetric matrices -- simple proofhttps://zbmath.org/1472.150432021-11-25T18:46:10.358925Z"Ferber, Asaf"https://zbmath.org/authors/?q=ai:ferber.asafSummary: In this paper we give a simple, short, and self-contained proof for a non-trivial upper bound on the probability that a random \(\pm 1\) symmetric matrix is singular.Some new results in random matrices over finite fieldshttps://zbmath.org/1472.150442021-11-25T18:46:10.358925Z"Luh, Kyle"https://zbmath.org/authors/?q=ai:luh.kyle"Meehan, Sean"https://zbmath.org/authors/?q=ai:meehan.sean"Nguyen, Hoi H."https://zbmath.org/authors/?q=ai:nguyen.hoi-hAuthors' abstract: In this note, we give various characterizations of random walks with possibly different steps that have relatively large discrepancy from the uniform distribution modulo a prime \(p\), and use these results to study the distribution of the rank of random matrices over \(\mathbb{F}_p\) and the equi-distribution behavior of normal vectors of random hyperplanes. We also study the probability that a random square matrix is eigenvalue-free, or when its characteristic polynomial is divisible by a given irreducible polynomial in the limit \(n \to \infty\) in \(\mathbb{F}_p\). We show that these statistics are universal, extending results of \textit{R. Stong} [Adv. Appl. Math. 9, No. 2, 167--199 (1988; Zbl 0681.05004)]
and \textit{P. M. Neumann} and \textit{C. E. Praeger} [J. Lond. Math. Soc., II. Ser. 58, No. 3, 564--586 (1999; Zbl 0936.15020); J. Algebra 234, No. 2, 367--418 (2000; Zbl 1020.20032)] beyond the uniform model.Some majorization inequalities induced by Schur products in Euclidean Jordan algebrashttps://zbmath.org/1472.170942021-11-25T18:46:10.358925Z"Gowda, M. Seetharama"https://zbmath.org/authors/?q=ai:gowda.muddappa-seetharamaSummary: In a Euclidean Jordan algebra \(\mathcal{V}\) of rank \(n\), an element \(x\) is said to be majorized by an element \(y\), symbolically \(x \prec y\), if the corresponding eigenvalue vector \(\lambda(x)\) is majorized by \(\lambda(y)\) in \(\mathcal{R}^n\). In this article, we describe pointwise majorization inequalities of the form \(T(x) \prec S(x)\), where \(T\) and \(S\) are linear transformations induced by Schur products. Specializing, we recover analogs of majorization inequalities of Schur, Hadamard, and Oppenheimer stated in the setting of Euclidean Jordan algebras, as well as majorization inequalities connecting quadratic and Lyapunov transformations on \(\mathcal{V} \). We also show how Schur products induced by certain scalar means (such as arithmetic, geometric, harmonic, and logarithmic means) naturally lead to majorization inequalities.Rota-Baxter operators on BiHom-associative algebras and related structureshttps://zbmath.org/1472.170972021-11-25T18:46:10.358925Z"Liu, Ling"https://zbmath.org/authors/?q=ai:liu.ling"Makhlouf, Abdenacer"https://zbmath.org/authors/?q=ai:makhlouf.abdenacer"Menini, Claudia"https://zbmath.org/authors/?q=ai:menini.claudia"Panaite, Florin"https://zbmath.org/authors/?q=ai:panaite.florinFollowing the work of Makhlouf and Yau on Rota-Baxter Hom-algebras, Rota-Baxter on BiHom associative algebras are introduced. A BiHom associative algebra is an associative algebra with two commuting algebra endomorphisms \(\alpha\) and \(\beta\), with the following axiom:
\[
\alpha(x)yz=xy\beta(z).
\]
Firstly, BiHom dendriform, Zinbiel, tridendriform and quadri algebras are introduced, and classical relations between these objects are extended to the BiHom context. It is also proved that a Yau twist exists for all of them. Secondly, a theory of Rota-Baxter operators on these objects is developed. It is proved that if the Rota-Baxter operator \(R\) commutes with both algebra endomorphisms, then the Yau twist is also a Rota-Baxter algebra. Free Rota-Baxter BiHom associative algebras are described with the help of planar trees and with considerations of functors related to Rota-Baxter structures. The paper ends with considerations on weak pseudotwistors.A short proof of the symmetric determinantal representation of polynomialshttps://zbmath.org/1472.320012021-11-25T18:46:10.358925Z"Stefan, Anthony"https://zbmath.org/authors/?q=ai:stefan.anthony"Welters, Aaron"https://zbmath.org/authors/?q=ai:welters.aaron-tA recent theorem [\textit{J. W. Helton} et al., J. Funct. Anal. 240, No. 1, 105--191 (2006; Zbl 1135.47005)] asserts that a real, multivariate polynomial can be written as the determinant of an affine pencil of real, symmetric matrices. The proof by Helton et al. was derived from an elaborate construct of non-commutative algebra, in its turn inspired by control system theory. The note by Stefan and Welters offers an elementary proof of the same result. The authors ingeniously exploit Schur complement identities, allowing a generalization of the main result to certain fields of finite characteristic. The strong algorithmic flavor of the proof may appeal to wider groups of scientists touching numerical matrix analysis in their studies.Commutativity of quaternion-matrix-valued functions and quaternion matrix dynamic equations on time scaleshttps://zbmath.org/1472.341642021-11-25T18:46:10.358925Z"Li, Zhien"https://zbmath.org/authors/?q=ai:li.zhien"Wang, Chao"https://zbmath.org/authors/?q=ai:wang.chao"Agarwal, Ravi P."https://zbmath.org/authors/?q=ai:agarwal.ravi-p"O'Regan, Donal"https://zbmath.org/authors/?q=ai:oregan.donalSummary: In this paper, we obtain some basic results of quaternion algorithms and quaternion calculus on time scales. Based on this, a Liouville formula and some related properties are derived for quaternion dynamic equations on time scales through conjugate transposed matrix algorithms. Moreover, we introduce the quaternion matrix exponential function by homogeneous quaternion matrix dynamic equations. Also a corresponding existence and uniqueness theorem is proved. In addition, the commutativity of \(n \times n\) quaternion-matrix-valued functions is investigated and some sufficient and necessary conditions of commutativity and noncommutativity are established on time scales. Also the fundamental solution matrices of some basic quaternion matrix dynamic equations are obtained. Examples are provided to illustrate the results, which are completely new on hybrid domains particularly when the time scales are the quantum case \(\mathbb{T}=\overline{q^{\mathbb{Z}}}\) and the discrete case \(\mathbb{T}=h\mathbb{Z}; h > 0\), both of which are significant for the study of quaternion q-dynamic equations and quaternion difference dynamic equations. Finally, we present several applications including multidimensional rotations and transformations of the submarine, the gyroscope, and the planet whose dynamical behaviors are depicted by quaternion dynamics on time scales and the corresponding iteration numerical solution for homogeneous quaternion dynamic equations are provided on various time scales.On differential equations derived from the pseudospherical surfaceshttps://zbmath.org/1472.353432021-11-25T18:46:10.358925Z"Yang, Hongwei"https://zbmath.org/authors/?q=ai:yang.hongwei"Wang, Xiangrong"https://zbmath.org/authors/?q=ai:wang.xiangrong"Yin, Baoshu"https://zbmath.org/authors/?q=ai:yin.baoshuSummary: We construct two metric tensor fields; by means of these metric tensor fields, sinh-Gordon equation and elliptic sinh-Gordon equation are obtained, which describe pseudospherical surfaces of constant negative Riemann curvature scalar \(\sigma\) = \(-\)2, \(\sigma\) = \(-\)1, respectively. By employing the Bäcklund transformation, nonlinear superposition formulas of sinh-Gordon equation and elliptic sinh-Gordon equation are derived; various new exact solutions of the equations are obtained.Painlevé IV and the semi-classical Laguerre unitary ensembles with one jump discontinuitieshttps://zbmath.org/1472.370622021-11-25T18:46:10.358925Z"Zhu, Mengkun"https://zbmath.org/authors/?q=ai:zhu.mengkun"Wang, Dan"https://zbmath.org/authors/?q=ai:wang.dan"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1|chen.yang.2Summary: In this paper, we present the characteristic of a certain discontinuous linear statistic of the semi-classical Laguerre unitary ensembles
\[
w(z,t)=A\theta (z-t)e^{-z^2+tz},
\]
here \(\theta(x)\) is the Heaviside function, where \(A> 0\), \(t>0\), and \(z\in [0,\infty)\). We derive the ladder operators and its interrelated compatibility conditions. By using the ladder operators, we show two auxiliary quantities \(R_n(t)\) and \(r_n(t)\) satisfy the coupled Riccati equations, from which we also prove that \(R_n(t)\) satisfies a particular Painlevé IV equation. Even more, \(\sigma_n(t)\), allied to \(R_n(t)\), satisfies both the discrete and continuous Jimbo-Miwa-Okamoto \(\sigma\)-form of the Painlevé IV equation. Finally, we consider the situation when \(n\) gets large, the second order linear differential equation satisfied by the polynomials \(P_n(x)\) orthogonal with respect to the semi-classical weight turn to be a particular bi-confluent Heun equation.A matrix approach to some second-order difference equations with sign-alternating coefficientshttps://zbmath.org/1472.390042021-11-25T18:46:10.358925Z"Anđelić, Milica"https://zbmath.org/authors/?q=ai:andelic.milica"Du, Zhibin"https://zbmath.org/authors/?q=ai:du.zhibin"da Fonseca, Carlos M."https://zbmath.org/authors/?q=ai:da-fonseca.carlos-martins"Kılıç, Emrah"https://zbmath.org/authors/?q=ai:kilic.emrahSummary: In this paper, we analyse and unify some recent results on the double sequence \(\{y_{n,k}\}\), for \(n, k\geqslant 1\), defined by the second-order difference equation \[y_{n,k}=(-1)^{\lfloor(n-1)/k\rfloor}y_{n-1,k}-y_{n-2,k},\]
with \(y_{1,k}=1\) and \(y_{2,k}=0\), in terms of matrix theory and orthogonal polynomials theory. Moreover, we provide a general solution to
\[z_{n,k}=(-1)^{\lfloor(n-1)/k\rfloor}z_{n-1,k}-(-1)^{\lfloor (n-2)/a\rfloor}z_{n-2,k},\]
using a closely related approach. We discuss briefly other recent problems involving a general recurrence relation of second order and relate them with the existing literature.An improved uncertainty principle for functions with symmetryhttps://zbmath.org/1472.430042021-11-25T18:46:10.358925Z"Garcia, Stephan Ramon"https://zbmath.org/authors/?q=ai:garcia.stephan-ramon"Karaali, Gizem"https://zbmath.org/authors/?q=ai:karaali.gizem"Katz, Daniel J."https://zbmath.org/authors/?q=ai:katz.daniel-jThe authors prove a generalization of a result by Chebotarëv which states that every minor of a discrete Fourier matrix of prime order is nonzero. A generalization of the Biró-Meshulam-Tao uncertainty principle
[\textit{R.~Meshulam}, Eur. J. Comb. 27, No.~1, 63--67 (2006; Zbl 1145.43005);
\textit{T.~Tao}, Math. Res. Lett. 12, No.~1, 121--127 (2005; Zbl 1080.42002)]
to functions with symmetries that arise from certain group actions is used to establish the result. As special cases, the result includes analogues for discrete cosine and discrete sine matrices. The authors also show that their result is best possible and in some cases is stronger than that of Biró-Meshulam-Tao. Some of these results are shown to hold for non-prime fields under certain conditions.On the support of the free additive convolutionhttps://zbmath.org/1472.460692021-11-25T18:46:10.358925Z"Bao, Zhigang"https://zbmath.org/authors/?q=ai:bao.zhigang"Erdős, László"https://zbmath.org/authors/?q=ai:erdos.laszlo"Schnelli, Kevin"https://zbmath.org/authors/?q=ai:schnelli.kevinSummary: We consider the free additive convolution of two probability measures \(\mu\) and \(\nu\) on the real line and show that \(\mu\boxplus v\) is supported on a single interval if \(\mu\) and \(\nu\) each has single interval support. Moreover, the density of \(\mu\boxplus\nu\) is proven to vanish as a square root near the edges of its support if both \(\mu\) and \(\nu\) have power law behavior with exponents between \(-1\) and 1 near their edges. In particular, these results show the ubiquity of the conditions in our recent work on optimal local law at the spectral edges for addition of random matrices [\textit{Z.-G. Bao} et al., J. Funct. Anal. 279, No. 7, Article ID 108639, 93~p. (2020; Zbl 1460.46058)].Asymptotic formulas for determinants of a special class of Toeplitz + Hankel matriceshttps://zbmath.org/1472.470212021-11-25T18:46:10.358925Z"Basor, Estelle"https://zbmath.org/authors/?q=ai:basor.estelle-l"Ehrhardt, Torsten"https://zbmath.org/authors/?q=ai:ehrhardt.torstenSummary: We compute the asymptotics of the determinants of certain \(n \times n\) Toeplitz + Hankel matrices \( T_{n}(a)+H_n(b)\) as \( n\rightarrow \infty \) with symbols of Fisher-Hartwig type. More specifically, we consider the case where \(a\) has zeros and poles and where \(b\) is related to \(a\) in specific ways. Previous results of \textit{P. Deift} et al. [Ann. Math. (2) 174, No. 2, 1243--1299 (2011; Zbl 1232.15006)] dealt with the case where \(a\) is even. We are generalizing this in a mild way to certain non-even symbols.
For the entire collection see [Zbl 1367.47005].The theory of generalized locally Toeplitz sequences: a review, an extension, and a few representative applicationshttps://zbmath.org/1472.470222021-11-25T18:46:10.358925Z"Garoni, Carlo"https://zbmath.org/authors/?q=ai:garoni.carlo"Serra-Capizzano, Stefano"https://zbmath.org/authors/?q=ai:serra-capizzano.stefanoSummary: We review and extend the theory of generalized locally Toeplitz (GLT) sequences, which goes back to \textit{P. Tilli}'s work on locally Toeplitz sequences [Linear Algebra Appl. 278, No. 1--3, 91--120 (1998; Zbl 0934.15009)] and was developed by the second author during the last decade. Informally speaking, a GLT sequence \(\{A_{n}\}_n\) is a sequence of matrices with increasing size equipped with a function \(\kappa\) (the so-called symbol). We write \(\{A_{n}\}_n \sim_{\text{\textsc{glt}}}\kappa\) to indicate that \(\{A_{n}\}_n\) is a GLT sequence with symbol \(\kappa\). This symbol characterizes the asymptotic singular value distribution of \(\{A_{n}\}_n\); if the matrices \(A_n\) are Hermitian, it also characterizes the asymptotic eigenvalue distribution of \(\{A_{n}\}_n\). Three fundamental examples of GLT sequences are: (i) the sequence of Toeplitz matrices generated by a function \(f\) in \(L^{1}\); (ii) the sequence of diagonal sampling matrices containing the samples of a Riemann-integrable function \(a\) over equispaced grids; (iii) any zero-distributed sequence, i.e., any sequence of matrices with an asymptotic singular value distribution characterized by 0. The symbol of the GLT sequence (i) is \(f\), the symbol of the GLT sequence (ii) is \(a\), and the symbol of the GLT sequences (iii) is \(0\). The set of GLT sequences is a \(^\ast\)-algebra. More precisely, suppose that \(\{A_{n}^{(i)}\}_{n} \sim_{\text{\textsc{glt}}}\kappa_i\) for \(i = 1,\dots ,r\), and let \(A_n = \mathrm{ops}(A_{n}^{(1)},\dots , A_{n}^{(r)})\) be a matrix obtained from \(A_{n}^{(1)},\dots , A_{n}^{(r)}\) by means of certain algebraic operations ``ops'', such as linear combinations, products, inversions and conjugate transpositions; then \(\{A_{n}\}_{n} \sim_{\text{\textsc{glt}}} \kappa = \mathrm{ops}(\kappa_{1},\dots , k_r)\).
The theory of GLT sequences is a powerful apparatus for computing the asymptotic singular value and eigenvalue distribution of the discretization matrices \(A_n\) arising from the numerical approximation of continuous problems, such as integral equations and, especially, partial differential equations. Indeed, when the discretization parameter \(n\) tends to infinity, the matrices \(A_n\) give rise to a sequence \(\{A_n\}_n\), which often turns out to be a GLT sequence.
Nevertheless, this work is not primarily concerned with the applicative interest of the theory of GLT sequences. Although we will provide some illustrative applications at the end, the attention is focused on the mathematical foundations of the theory. We first propose a modification of the original definition of GLT sequences. With the new definition, we are able to enlarge the applicability of the theory, by generalizing/simplifying a lot of key results. In particular, we remove the Riemann-integrability assumption from the main spectral distribution and algebraic results for GLT sequences. As a final step, we extend the theory. We first prove an approximation result, which is useful to show that a given sequence of matrices is a GLT sequence. By using this result, we provide a new and easier proof of the fact that \(\{A_n^{-1}\}_n \sim _{\text{\textsc{glt}}}\kappa^{-1}\) whenever \(\{A_n\}_n \sim {\text{\textsc{glt}}}\, \kappa\), the matrices \(A_n\) are invertible, and \(\kappa\neq 0\) almost everywhere. Finally, using again the approximation result, we prove that \(\{f(A_{n})\}_n \sim {\text{\textsc{glt}}}f(\kappa)\) whenever \(\{A_n\}_n \sim_{\text{\textsc{glt}}}\kappa\), the matrices \(A_n\) are Hermitian, and \(f : \mathbb{R}\to \mathbb{R}\) is continuous.
For the entire collection see [Zbl 1367.47005].The circular law for sparse non-Hermitian matriceshttps://zbmath.org/1472.600062021-11-25T18:46:10.358925Z"Basak, Anirban"https://zbmath.org/authors/?q=ai:basak.anirban"Rudelson, Mark"https://zbmath.org/authors/?q=ai:rudelson.markLet \(\lambda_1\),\dots, \(\lambda_n\) the eigenvalues of a \(n\times n\) matrix \(B\); its empirical spectral distribution (ESD) is defined by \(L_B:=\frac{1}{n}\,\sum_{i=1}^n \delta_{\lambda_i}\), where \(\delta_x\) is the Dirac measure concentrated at \(x\). The sub-Gausssian norm of a random variable \(\xi\) is defined by
\[
\|\xi\|_{\psi_2}:=\sup_{k\ge 1} k^{-1/2}\,\mathbb{E}^{1/k}(|\xi|^k)\,.
\]
The main result of this important paper is the following theorem, which extends previous results quoted in the introduction.
Theorem. Let \(A_n\) be an \(n\times n\) matrix with i.i.d. entries \(a_{i,j}=\delta_{i,j}\,\xi_{i,j}\), where the \(\delta_{i,j}\) are independent Bernoulli random variables taking the value \(1\) with probability \(p_n\in\left]0,1\right]\) and \(\xi_{i,j}\) are real-valued i.i.d. sub-Gaussian centred random variables with unit variance.
\begin{enumerate}
\item[(i)] If \(p_n\) is such that \(np_n=\omega(\log^2 n)\), then as \(n\to\infty\) the \textnormal{ESD} of \(A_n/\sqrt{n\,p_n}\) converges weakly in probability to the circular law.
\item[(ii)] There exists a constant \(c\), which depends only on the sub-Gaussian norm of \(\{\xi_{i,j}\}\), such that if \(p_n\) satisfies the inequality \(np_n>\exp(c\,\sqrt{\log n})\), then the conclusion of (i) holds almost surely.
\end{enumerate}
Here, if \((a_n)\) and \((b_n)\) are two sequences of positive reals, one writes \(a_n=\omega(b_n)\) if \(b_n=o(a_n)\), \(a_n=O(b_n)\) and \(\limsup_{n\to\infty} a_n/b_n<\infty\)Traces of powers of matrices over finite fieldshttps://zbmath.org/1472.600082021-11-25T18:46:10.358925Z"Gorodetsky, Ofir"https://zbmath.org/authors/?q=ai:gorodetsky.ofir"Rodgers, Brad"https://zbmath.org/authors/?q=ai:rodgers.bradThe authors consider a prime power \(q=p^r,\) a matrix \(M\) chosen uniformly from the finite unitary group \(\mathrm{U}(n,q)\subset \mathrm{GL}(n,q^2),\) and the sequence \(\{M^i\}_{1\leq i \leq k}\) where \(i\) is not multiple of \(p.\) They prove that the traces of powers of matrices converge to independent uniform variables in \(\mathbb F_{q^2}\) as \(n \rightarrow \infty.\) The rate of convergence is shown to be exponential in \(n^2.\) \newline The related problem of the rate at which characteristic polynomial of \(M\) equidistributes in `short intervals' of \(\mathbb F_{q^2} [T]\) is also considered. \newline Analogous results are also proven for the general linear, special linear, symplectic and orthogonal groups over a finite field. \newline The proofs depend upon applying techniques from analytic number theory over function fields to formulas due to Fulman and others for the probability that the characteristic polynomial of a random matrix equals a given polynomial.Local laws for non-Hermitian random matrices and their productshttps://zbmath.org/1472.600092021-11-25T18:46:10.358925Z"Götze, Friedrich"https://zbmath.org/authors/?q=ai:gotze.friedrich-w"Naumov, Alexey"https://zbmath.org/authors/?q=ai:naumov.a-a"Tikhomirov, Alexander"https://zbmath.org/authors/?q=ai:tikhomirov.alexander-nThe smallest eigenvalue distribution of the Jacobi unitary ensembleshttps://zbmath.org/1472.600112021-11-25T18:46:10.358925Z"Lyu, Shulin"https://zbmath.org/authors/?q=ai:lyu.shulin"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1Summary: In the hard edge scaling limit of the Jacobi unitary ensemble generated by the weight \(x^{\alpha}(1 - x)^{\beta}, x \in [0, 1], \alpha, \beta > -1\), the probability that all eigenvalues of Hermitian matrices from this ensemble lie in the interval \([t, 1]\) is given by the Fredholm determinant of the Bessel kernel. We derive the constant in the asymptotics of this Bessel kernel determinant. A specialization of the results gives the constant in the asymptotics of the probability that the interval \((- a, a), a > 0\) is free of eigenvalues in the Jacobi unitary ensemble with the symmetric weight \((1 - x^2)^{\beta}, x \in [- 1, 1]\).Large deviations for extreme eigenvalues of deformed Wigner random matriceshttps://zbmath.org/1472.600122021-11-25T18:46:10.358925Z"Mckenna, Benjamin"https://zbmath.org/authors/?q=ai:mckenna.benjaminThe purpose of the paper is to prove a large deviation principle (LDP) for the largest eigenvalue of the random matrix \({X_N} = \frac{{{W_N}}}{{\sqrt N }} + {D_N}\), where \(\frac{{{W_N}}}{{\sqrt N }}\) lies in a particular class of real or complex Wigner matrices. In particular this includes Gaussian ensembles with full-rank general deformation. For the non-Gaussian ensembles, the deformation should be diagonal, and the laws of the entries of \({W_N}\) are supposed to have sharp sub-Gaussian Laplace transforms and satisfy certain concentration properties. It is also assumed that \({D_N}\) is a deterministic matrix whose empirical spectral measure tends to a deterministic limit \({\mu _D}\) and whose extreme eigenvalues tend to the edges of \({\mu _D}\). For these ensembles the paper establishes LDP in a restricted range \(( - \infty ,{x_c})\), where \({x_c}\) depends on the deformation only and can be infinite.Concentration inequalities for random tensorshttps://zbmath.org/1472.600412021-11-25T18:46:10.358925Z"Vershynin, Roman"https://zbmath.org/authors/?q=ai:vershynin.romanLet \(x_1,x_2,\ldots\) be independent random vectors in \(\mathbb{R}^n\) whose coordinates are independent random variables with zero mean and unit variance, and let \(X=x_1\otimes\cdots\otimes x_d\), a random tensor in \(\mathbb{R}^{n^d}\). The author proves two concentration inequalities for \(X\). Firstly, in the case where the \(x_k\) are bounded almost surely, it is shown that for a convex and Lipschitz function \(f\), and all \(0\leq t\leq2(\mathbb{E}|f(X)|^2)^{1/2}\), we have
\[
\mathbb{P}\left(\big|f(X)-\mathbb{E}f(X)\big|>t\right)\leq2\exp\left(-\frac{ct^2}{dn^{d-1}\|f\|^2_{Lip}}\right)\,,
\]
for some constant \(c>0\) depending on the bound for the \(x_k\). Secondly, in the case where the \(x_k\) are sub-Gaussian, it is shown that for a linear operator \(A\) taking values in a Hilbert space \(H\), and all \(0\leq t\leq2\|A\|_{HS}\), we have
\[
\mathbb{P}\left(\big|\|AX\|_H-\|A\|_{HS}\big|\geq t\right)\leq2\exp\left(-\frac{ct^2}{dn^{d-1}\|A\|^2_{op}}\right)\,,
\]
where \(c>0\) again depends on the \(x_k\), and where \(\|A\|_{HS}\) and \(\|A\|_{op}\) are the Hilbert-Schmidt and operator norms of \(A\), respectively. As an application of this latter concentration bound, the author shows that random tensors are well conditioned; that is, if \(d=o(\sqrt{n/\log(n)})\) then with high probability \((1-o(1))n^d\) independent copies of \(X\) are far from linearly dependent.Variance Laplacian: quadratic forms in statisticshttps://zbmath.org/1472.620792021-11-25T18:46:10.358925Z"Murthy, Garimella Rama"https://zbmath.org/authors/?q=ai:murthy.garimella-ramaSummary: In this research paper, it is proved that the variance of a discrete random variable, \(Z\) can be expressed as a quadratic form associated with a Laplacian matrix i.e.
\[
\mathrm{Variance }[Z]=X^T G X
\]
\(G\) is Laplacian matrix whose elements are expressed in terms of probabilities. We formally state and prove the properties of Variance Laplacian matrix, \(G\). Some implications of the properties of such matrix to statistics are discussed. It is reasoned that several interesting quadratic forms can be naturally associated with statistical measures such as the covariance of two random variables. It is hoped that VARIANCE LAPLACIAN MATRIX \(G\) will be of significant interest in statistical applications. The results are generalized to continuous random variables also. It is reasoned that cross-fertilization of results from the theory of quadratic forms and probability theory/statistics will lead to new research directions.
For the entire collection see [Zbl 1468.60003].Modified Mahalanobis-Taguchi system based on proper orthogonal decomposition for high-dimensional-small-sample-size data classificationhttps://zbmath.org/1472.620992021-11-25T18:46:10.358925Z"Mao, Ting"https://zbmath.org/authors/?q=ai:mao.ting"Yu, Lanting"https://zbmath.org/authors/?q=ai:yu.lanting"Zhang, Yueyi"https://zbmath.org/authors/?q=ai:zhang.yueyi"Zhou, Li"https://zbmath.org/authors/?q=ai:zhou.liSummary: Mahalanobis-Taguchi System (MTS) is an effective algorithm for dimensionality reduction, feature extraction and classification of data in a multidimensional system. However, when applied to the field of high-dimensional small sample data, MTS has challenges in calculating the Mahalanobis distance due to the singularity of the covariance matrix. To this end, we construct a modified Mahalanobis-Taguchi System (MMTS) by introducing the idea of proper orthogonal decomposition (POD). The constructed MMTS expands the application scope of MTS, taking into account correlations between variables and the influence of dimensionality. It can not only retain most of the original sample information features, but also achieve a substantial reduction in dimensionality, showing excellent classification performance. The results show that, compared with expert classification, individual classifiers such as NB, RF, k-NN, SVM and superimposed classifiers such as Wrapper + RF, MRMR + SVM, Chi-square + BP, SMOTE + Wrapper + RF and SMOTE + MRMR + SVM, MMTS has a better classification performance when extracting orthogonal decomposition vectors with eigenvalues greater than 0.001.On maximum residual block and two-step Gauss-Seidel algorithms for linear least-squares problemshttps://zbmath.org/1472.650402021-11-25T18:46:10.358925Z"Liu, Yong"https://zbmath.org/authors/?q=ai:liu.yong.2|liu.yong.4|liu.yong.3|liu.yong.5|liu.yong.1"Jiang, Xiang-Long"https://zbmath.org/authors/?q=ai:jiang.xianglong"Gu, Chuan-Qing"https://zbmath.org/authors/?q=ai:gu.chuanqingSummary: The block Gauss-Seidel algorithm can significantly outperform the simple randomized Gauss-Seidel algorithm for solving overdetermined least-squares problems since it moves a large block of columns rather than a single column into working memory. Here, with the help of the maximum residual rule, we construct a two-step Gauss-Seidel (2SGS) algorithm, which selects two different columns simultaneously at each iteration. As a natural extension of the 2SGS algorithm, we further propose a multi-step Gauss-Seidel algorithm, that is, the maximum residual block Gauss-Seidel (MRBGS) algorithm for solving overdetermined least-squares problems. We prove that these two different algorithms converge to the unique solution of the overdetermined least-squares problem when its coefficient matrix is of full column rank. Numerical experiments on Gaussian models as well as on 2D image reconstruction problems, show that 2SGS is more effective than the greedy randomized Gauss-Seidel algorithm, and MRBGS apparently outperforms both the greedy and randomized block Gauss-Seidel algorithms in terms of numerical performance.A framework for second-order eigenvector centralities and clustering coefficientshttps://zbmath.org/1472.650432021-11-25T18:46:10.358925Z"Arrigo, Francesca"https://zbmath.org/authors/?q=ai:arrigo.francesca"Higham, Desmond J."https://zbmath.org/authors/?q=ai:higham.desmond-j"Tudisco, Francesco"https://zbmath.org/authors/?q=ai:tudisco.francescoSummary: We propose and analyse a general tensor-based framework for incorporating second-order features into network measures. This approach allows us to combine traditional pairwise links with information that records whether triples of nodes are involved in wedges or triangles. Our treatment covers classical spectral methods and recently proposed cases from the literature, but we also identify many interesting extensions. In particular, we define a mutually reinforcing (spectral) version of the classical clustering coefficient. The underlying object of study is a constrained nonlinear eigenvalue problem associated with a cubic tensor. Using recent results from nonlinear Perron-Frobenius theory, we establish existence and uniqueness under appropriate conditions, and show that the new spectral measures can be computed efficiently with a nonlinear power method. To illustrate the added value of the new formulation, we analyse the measures on a class of synthetic networks. We also give computational results on centrality and link prediction for real-world networks.Computing tensor Z-eigenvalues via shifted inverse power methodhttps://zbmath.org/1472.650462021-11-25T18:46:10.358925Z"Sheng, Zhou"https://zbmath.org/authors/?q=ai:sheng.zhou"Ni, Qin"https://zbmath.org/authors/?q=ai:ni.qinSummary: The positive definiteness of an even degree homogeneous polynomial plays an important role in the stability study of nonlinear autonomous systems via Lyapunov's direct method in automatic control, the detection of \(\mathcal{P}\) or \(\mathcal{P}_0\) tensor in tensor complementarity problems and spectral hypergraph theory, and more. Owing to the positive definiteness of an even degree homogeneous polynomial is equivalent to that of an even order symmetric tensor. In this paper, we propose a shifted inverse power method for computing tensor Z-eigenpairs, which can be viewed as a generalization of the inverse power method for matrices case. We also formulate it as a fixed point iteration form, and reveal that the relationship between the fixed points and the Z-eigenvectors of symmetric tensors. The advantages of the proposed method are simple operations and readily comprehensible convergence analysis. An efficient initialization strategy is also developed, which makes the proposed method converges to a better solution compared to not using the initialization strategy. Finally, we present applications of the proposed method in nonlinear autonomous systems, the detection of \(\mathcal{P}\) or \(\mathcal{P}_0\) tensor and symmetric tensor Z-eigenproblems, some numerical results are reported to illustrate the effectiveness of the proposed method.An algorithm based on QSVD for the quaternion equality constrained least squares problemhttps://zbmath.org/1472.650482021-11-25T18:46:10.358925Z"Zhang, Yanzhen"https://zbmath.org/authors/?q=ai:zhang.yanzhen"Li, Ying"https://zbmath.org/authors/?q=ai:li.ying.1"Wei, Musheng"https://zbmath.org/authors/?q=ai:wei.musheng"Zhao, Hong"https://zbmath.org/authors/?q=ai:zhao.hongSummary: Quaternion equality constrained least squares (QLSE) problems have attracted extensive attention in the field of mathematical physics due to its applicability as an extremely effective tool. However, the knowledge gap among numerous QLSE problems has not been settled now. In this paper, by using quaternion SVD (Q-SVD) and the equivalence of the QLSE problem and Karush-Kuhb-Tucker (KKT) equation, we obtain some equations about the matrices in the general solution of the QLSE problem. Using these equations, an equivalent form of the solution of the QLSE problem is obtained. Then, applying the special structure of real representation of quaternion, we propose a real structure-preserving algorithm based on Q-SVD. At last, we give numerical example, which illustrates the effectiveness of our algorithm.Tensor extrapolation methods with applicationshttps://zbmath.org/1472.650492021-11-25T18:46:10.358925Z"Beik, F. P. A."https://zbmath.org/authors/?q=ai:beik.fatemeh-panjeh-ali"El Ichi, A."https://zbmath.org/authors/?q=ai:el-ichi.alaa"Jbilou, K."https://zbmath.org/authors/?q=ai:jbilou.khalide"Sadaka, R."https://zbmath.org/authors/?q=ai:sadaka.rachidSummary: In this paper, we mainly develop the well-known vector and matrix polynomial extrapolation methods in tensor framework. To this end, some new products between tensors are defined and the concept of positive definitiveness is extended for tensors corresponding to T-product. Furthermore, we discuss on the solution of least-squares problem associated with a tensor equation using Tensor Singular Value Decomposition (TSVD). Motivated by the effectiveness of some proposed vector extrapolation methods in earlier papers, we describe how an extrapolation technique can be also implemented on the sequence of tensors produced by truncated TSVD (TTSVD) for solving possibly ill-posed tensor equations.A block diagonalization based algorithm for the determinants of block \(k\)-tridiagonal matriceshttps://zbmath.org/1472.650502021-11-25T18:46:10.358925Z"Jia, Ji-Teng"https://zbmath.org/authors/?q=ai:jia.jiteng"Yan, Yu-Cong"https://zbmath.org/authors/?q=ai:yan.yu-cong"He, Qi"https://zbmath.org/authors/?q=ai:he.qiSummary: In the current paper, we present a numerical algorithm for computing the determinants of block \(k\)-tridiagonal matrices. The algorithm is based on the use of a fast block diagonalization method and any algorithm for evaluating block tridiagonal determinants. Meanwhile, an explicit numerical formula for the block \(k\)-tridiagonal determinants is also derived, which is based on the combination of the proposed block diagonalization method and a two-term recurrence for block tridiagonal determinants. The experimental results of some representative numerical examples are provided to show the validity and effectiveness of the proposed algorithm and its competitiveness with MATLAB built-in function.Performance and stability of direct methods for computing generalized inverses of the graph Laplacianhttps://zbmath.org/1472.650522021-11-25T18:46:10.358925Z"Benzi, Michele"https://zbmath.org/authors/?q=ai:benzi.michele"Fika, Paraskevi"https://zbmath.org/authors/?q=ai:fika.paraskevi"Mitrouli, Marilena"https://zbmath.org/authors/?q=ai:mitrouli.marilenaThe authors present and study two direct algorithms for computing particular generalized inverses of graph Laplacians: the group inverse and the absorption inverse. The group generalized inverse of a square matrix \(A\) is the unique \(A^\sharp\) such that \(AA^\sharp A=A\), \(A^\sharp AA^\sharp=A^\sharp\) and \(AA^\sharp=A^\sharp A\), while absorption inverse further generalizes group inverse for graphs with absorption. The authors first improve the direct method they previously developed in [Linear Algebra Appl. 574, 123--158 (2019; Zbl. 1433.65065)], and show that this improvement is forward stable with high relative component-wise accuracy. The second proposed algorithm is based on bottleneck matrices, which represents inverses of the principal \((n-1)\times(n-1)\)-minors of the Laplacian, with lower numerical stability than the first method. Experimental results show that both methods presented here provide faster algorithms for handling dense problems.A note on nonclosed tensor formatshttps://zbmath.org/1472.650532021-11-25T18:46:10.358925Z"Hackbusch, W."https://zbmath.org/authors/?q=ai:hackbusch.wolfgangSummary: Various tensor formats exist which allow a data-sparse representation of tensors. Some of these formats are not closed. The consequences are (i) possible non-existence of best approximations and (ii) divergence of the representing parameters when a tensor within the format tends to a border tensor outside. The paper tries to describe the nature of this divergence. A particular question is whether the divergence is uniform for all border tensors.Algorithms for nonnegative matrix factorization with the Kullback-Leibler divergencehttps://zbmath.org/1472.650542021-11-25T18:46:10.358925Z"Hien, Le Thi Khanh"https://zbmath.org/authors/?q=ai:hien.le-thi-khanh"Gillis, Nicolas"https://zbmath.org/authors/?q=ai:gillis.nicolasSummary: Nonnegative matrix factorization (NMF) is a standard linear dimensionality reduction technique for nonnegative data sets. In order to measure the discrepancy between the input data and the low-rank approximation, the Kullback-Leibler (KL) divergence is one of the most widely used objective function for NMF. It corresponds to the maximum likehood estimator when the underlying statistics of the observed data sample follows a Poisson distribution, and KL NMF is particularly meaningful for count data sets, such as documents. In this paper, we first collect important properties of the KL objective function that are essential to study the convergence of KL NMF algorithms. Second, together with reviewing existing algorithms for solving KL NMF, we propose three new algorithms that guarantee the non-increasingness of the objective function. We also provide a global convergence guarantee for one of our proposed algorithms. Finally, we conduct extensive numerical experiments to provide a comprehensive picture of the performances of the KL NMF algorithms.Nonconvex optimization for robust tensor completion from grossly sparse observationshttps://zbmath.org/1472.650562021-11-25T18:46:10.358925Z"Zhao, Xueying"https://zbmath.org/authors/?q=ai:zhao.xueying"Bai, Minru"https://zbmath.org/authors/?q=ai:bai.minru"Ng, Michael K."https://zbmath.org/authors/?q=ai:ng.michael-kSummary: In this paper, we consider the robust tensor completion problem for recovering a low-rank tensor from limited samples and sparsely corrupted observations, especially by impulse noise. A convex relaxation of this problem is to minimize a weighted combination of tubal nuclear norm and the \(\ell_1\)-norm data fidelity term. However, the \(\ell_1\)-norm may yield biased estimators and fail to achieve the best estimation performance. To overcome this disadvantage, we propose and develop a nonconvex model, which minimizes a weighted combination of tubal nuclear norm, the \(\ell_1\)-norm data fidelity term, and a concave smooth correction term. Further, we present a Gauss-Seidel difference of convex functions algorithm (GS-DCA) to solve the resulting optimization model by using a linearization technique. We prove that the iteration sequence generated by GS-DCA converges to the critical point of the proposed model. Furthermore, we propose an extrapolation technique of GS-DCA to improve the performance of the GS-DCA. Numerical experiments for color images, hyperspectral images, magnetic resonance imaging images and videos demonstrate that the effectiveness of the proposed method.Newton's method for M-tensor equationshttps://zbmath.org/1472.650592021-11-25T18:46:10.358925Z"Li, Dong-Hui"https://zbmath.org/authors/?q=ai:li.donghui|li.dong-hui"Xu, Jie-Feng"https://zbmath.org/authors/?q=ai:xu.jiefeng"Guan, Hong-Bo"https://zbmath.org/authors/?q=ai:guan.hongboSummary: We are concerned with the tensor equations whose coefficient tensors are M-tensors. We first propose a Newton method for solving the equation with a positive constant term and establish its global and quadratic convergence. Then we extend the method to solve the equation with a nonnegative constant term and establish its convergence. At last, we do numerical experiments to test the proposed methods. The results show that the proposed methods are quite efficient.Efficient nonnegative matrix factorization by DC programming and DCAhttps://zbmath.org/1472.650702021-11-25T18:46:10.358925Z"Le Thi, Hoai An"https://zbmath.org/authors/?q=ai:le-thi-hoai-an."Vo, Xuan Thanh"https://zbmath.org/authors/?q=ai:vo.xuan-thanh"Dinh, Tao Pham"https://zbmath.org/authors/?q=ai:pham-dinh-tao.Summary: In this letter, we consider the nonnegative matrix factorization (NMF) problem and several NMF variants. Two approaches based on DC (difference of convex functions) programming and DCA (DC algorithm) are developed. The first approach follows the alternating framework that requires solving, at each iteration, two nonnegativity-constrained least squares subproblems for which DCA-based schemes are investigated. The convergence property of the proposed algorithm is carefully studied. We show that with suitable DC decompositions, our algorithm generates most of the standard methods for the NMF problem. The second approach directly applies DCA on the whole NMF problem. Two algorithms -- one computing all variables and one deploying a variable selection strategy -- are proposed. The proposed methods are then adapted to solve various NMF variants, including the nonnegative factorization, the smooth regularization NMF, the sparse regularization NMF, the multilayer NMF, the convex/convex-hull NMF, and the symmetric NMF. We also show that our algorithms include several existing methods for these NMF variants as special versions. The efficiency of the proposed approaches is empirically demonstrated on both real-world and synthetic data sets. It turns out that our algorithms compete favorably with five state-of-the-art alternating nonnegative least squares algorithms.Multi-parametric classification of automaton Markov models based on the sequences they generatehttps://zbmath.org/1472.681072021-11-25T18:46:10.358925Z"Nurutdinova, A. R."https://zbmath.org/authors/?q=ai:nurutdinova.a-r"Shalagin, S. V."https://zbmath.org/authors/?q=ai:shalagin.sergei-viktorovichSummary: This article is devoted to multi-parametric classification of automaton Markov models (AMMs) on the base of output sequences with the use of discriminant analysis. The AMMs under consideration are specified by means of stochastic matrices belonging to subclasses defined a priori. A set of claasification features is introduced to distinguish AMMs specified by matrices from different subclasses. The features are related to the frequency characteristics of sequences generated by AMMs. A method is suggested for determining the minimal length of the sequence need to calculate the features with a required accuracy.Convex coupled matrix and tensor completionhttps://zbmath.org/1472.681642021-11-25T18:46:10.358925Z"Wimalawarne, Kishan"https://zbmath.org/authors/?q=ai:wimalawarne.kishan"Yamada, Makoto"https://zbmath.org/authors/?q=ai:yamada.makoto"Mamitsuka, Hiroshi"https://zbmath.org/authors/?q=ai:mamitsuka.hiroshiSummary: We propose a set of convex low-rank inducing norms for coupled matrices and tensors (hereafter referred to as coupled tensors), in which information is shared between the matrices and tensors through common modes. More specifically, we first propose a mixture of the overlapped trace norm and the latent norms with the matrix trace norm, and then, propose a completion model regularized using these norms to impute coupled tensors. A key advantage of the proposed norms is that they are convex and can be used to find a globally optimal solution, whereas existing methods for coupled learning are nonconvex. We also analyze the excess risk bounds of the completion model regularized using our proposed norms and show that they can exploit the low-rankness of coupled tensors, leading to better bounds compared to those obtained using uncoupled norms. Through synthetic and real-data experiments, we show that the proposed completion model compares favorably with existing ones.The field of values of Jones matrices: classification and special caseshttps://zbmath.org/1472.780052021-11-25T18:46:10.358925Z"Gutiérrez-Vega, Julio C."https://zbmath.org/authors/?q=ai:gutierrez-vega.julio-cSummary: The concept of field of values (FoV), also known as the numerical range, is applied to the \(2 \times 2\) Jones matrices used in polarization optics. We discover the relevant interplay between the geometric properties of the FoV, the algebraic properties of the Jones matrices and the representation of polarization states on the Poincaré sphere. The properties of the FoV reveal hidden symmetries in the relationships between the eigenvectors and eigenvalues of the Jones matrices. We determine the main mathematical properties of the FoV, discuss the special cases that are relevant to polarization optics, and describe its application to calculate the Pancharatnam-Berry phase introduced by an optical system to the input state.Entanglement breaking channels, stochastic matrices, and primitivityhttps://zbmath.org/1472.810252021-11-25T18:46:10.358925Z"Ahiable, Jennifer"https://zbmath.org/authors/?q=ai:ahiable.jennifer"Kribs, David W."https://zbmath.org/authors/?q=ai:kribs.david-w"Levick, Jeremy"https://zbmath.org/authors/?q=ai:levick.jeremy"Pereira, Rajesh"https://zbmath.org/authors/?q=ai:pereira.rajesh"Rahaman, Mizanur"https://zbmath.org/authors/?q=ai:rahaman.mizanurSummary: We consider the important class of quantum operations (completely positive trace-preserving maps) called entanglement breaking channels. We show how every such channel induces stochastic matrix representations that have the same non-zero spectrum as the channel. We then use this to investigate when entanglement breaking channels are primitive, and prove this depends on primitivity of the matrix representations. This in turn leads to tight bounds on the primitivity index of entanglement breaking channels in terms of the primitivity index of the associated stochastic matrices. We also present examples and discuss open problems generated by the work.Experimental detection of qubit-ququart pseudo-bound entanglement using three nuclear spinshttps://zbmath.org/1472.810352021-11-25T18:46:10.358925Z"Singh, Amandeep"https://zbmath.org/authors/?q=ai:singh.amandeep"Gautam, Akanksha"https://zbmath.org/authors/?q=ai:gautam.akanksha"Arvind"https://zbmath.org/authors/?q=ai:arvind.vikraman|arvind.d-k|arvind.b|arvind.a|arvind.n|arvind.kim-p-gostelow|arvind.m-t"Dorai, Kavita"https://zbmath.org/authors/?q=ai:dorai.kavitaSummary: In this work, we experimentally created and characterized a class of qubit-ququart PPT (positive under partial transpose) entangled states using three nuclear spins on an nuclear magnetic resonance (NMR) quantum information processor. Entanglement detection and characterization for systems with a Hilbert space dimension \(\geq 2 \otimes 3\) is nontrivial since there are states in such systems which are both PPT as well as entangled. The experimental detection scheme that we devised for the detection of qubit-ququart PPT entanglement was based on the measurement of three Pauli operators with high precision, and is a key ingredient of the protocol in detecting entanglement. The family of PPT-entangled states considered in the current study are incoherent mixtures of five pure states. All the five states were prepared with high fidelities and the resulting PPT entangled states were prepared with mean fidelity \(\geq 0.95\). The entanglement thus detected was validated by carrying out full quantum state tomography (QST).Mutually unbiased bases containing a complex Hadamard matrix of Schmidt rank threehttps://zbmath.org/1472.810492021-11-25T18:46:10.358925Z"Hu, Mengyao"https://zbmath.org/authors/?q=ai:hu.mengyao"Chen, Lin"https://zbmath.org/authors/?q=ai:chen.lin.1|chen.lin.6|chen.lin.4|chen.lin.2|chen.lin.5|chen.lin|chen.lin.3"Sun, Yize"https://zbmath.org/authors/?q=ai:sun.yizeSummary: Constructing four six-dimensional mutually unbiased bases (MUBs) is an open problem in quantum physics and measurement. We investigate the existence of four MUBs including the identity, and a complex Hadamard matrix (CHM) of Schmidt rank three. The CHM is equivalent to a controlled unitary operation on the qubit-qutrit system via local unitary transformation \(I_2 \otimes V\) and \(I_2 \otimes W\). We show that \(V\) and \(W\) have no zero entry, and apply it to exclude constructed examples as members of MUBs. We further show that the maximum of entangling power of controlled unitary operation is log \(_2 3\) ebits. We derive the condition under which the maximum is achieved, and construct concrete examples. Our results describe the phenomenon that if a CHM of Schmidt rank three belongs to an MUB then its entangling power may not reach the maximum.MMSE detection method in uplink massive MIMO systems based on quantum computinghttps://zbmath.org/1472.810572021-11-25T18:46:10.358925Z"Wang, XiaoJun"https://zbmath.org/authors/?q=ai:wang.xiaojun.1"Zhao, Jie"https://zbmath.org/authors/?q=ai:zhao.jie"Meng, FanXu"https://zbmath.org/authors/?q=ai:meng.fanxu"Yu, XuTao"https://zbmath.org/authors/?q=ai:yu.xutao"Zhang, ZaiChen"https://zbmath.org/authors/?q=ai:zhang.zaichenSummary: The minimum mean square error (MMSE) detection method involved matrix inversion operation with excessive computational burden. In this paper, we develop an improved quantum linear system algorithm to solve matrix inversion problem of the MMSE detection method in uplink massive multiple-input and multiple-output (MIMO) systems. In order to apply reasonably the robust computational power of quantum computing, we optimize the preparation of the input state and the extraction of the solution from the final entangled quantum state. We prove that this algorithm can reduce computational complexity to \(\mathrm{O}(N \log N)\).Supertime and Pauli's principlehttps://zbmath.org/1472.811082021-11-25T18:46:10.358925Z"Musin, Y. R."https://zbmath.org/authors/?q=ai:musin.yu-rSummary: Connection of Pauli's principle with the nontrivial structure of the fermion supertime is shown. When supersymmetry is localized as supergravitation, fields of gravitational and exchange interaction carriers arise. The exchange interaction quantum of free fermions, being a superpartner of graviton (gravitino), is interpreted as a paulino -- the particle responsible for the effect of mutual avoidance of identical fermions.Axial momentum for the relativistic Majorana particlehttps://zbmath.org/1472.811182021-11-25T18:46:10.358925Z"Arodź, H."https://zbmath.org/authors/?q=ai:arodz.henrykSummary: The Hilbert space of states of the relativistic Majorana particle consists of normalizable bispinors with real components, hence the usual momentum operator \(- i\nabla\) can not be defined in this space. For this reason, we introduce the axial momentum operator, \(-i \gamma_5 \nabla\) as a new observable for this particle. In the Heisenberg picture, the axial momentum contains a component which oscillates with the amplitude proportional to \(m/E\), where \(E\) is the energy and \(m\) the mass of the particle. The presence of the oscillations discriminates between the massive and massless Majorana particle. Furthermore, we show how the eigenvectors of the axial momentum, called the axial plane waves, can be used as a basis for obtaining the general solution of the evolution equation, also in the case of free Majorana field. Here a novel feature is a coupling of modes with the opposite momenta, again present only in the case of massive particle or field.Further investigation of mass dimension one fermionic dualshttps://zbmath.org/1472.811192021-11-25T18:46:10.358925Z"Hoff da Silva, J. M."https://zbmath.org/authors/?q=ai:hoff-da-silva.julio-marny"Cavalcanti, R. T."https://zbmath.org/authors/?q=ai:cavalcanti.rogerio-t|cavalcanti.rafael-tSummary: In this paper we proceed into the next step of formalization of a consistent dual theory for mass dimension one spinors. This task is developed approaching the two different and complementary aspects of such duals, clarifying its algebraic structure and the so called \(\tau\)-deformation. The former regards the mathematical equivalence of the recent proposed Lorentz preserving dual with the duals of algebraic spinors, from Clifford algebras, showing the consistency and generality of the new dual. Moreover, by revealing its automorphism structure, the hole of the \(\tau\)-deformation and contrasting the action group orbits with other Lorentz breaking scenarios, we argue that the new mass dimension one dual theory is placed over solid and consistent basis.Spinorial \(R\) operator and algebraic Bethe ansatzhttps://zbmath.org/1472.811202021-11-25T18:46:10.358925Z"Karakhanyan, D."https://zbmath.org/authors/?q=ai:karakhanyan.david"Kirschner, R."https://zbmath.org/authors/?q=ai:kirschner.rolandSummary: We propose a new approach to the spinor-spinor R-matrix with orthogonal and symplectic symmetry. Based on this approach and the fusion method we relate the spinor-vector and vector-vector monodromy matrices for quantum spin chains. We consider the explicit spinor \(R\) matrices of low rank orthogonal algebras and the corresponding \(RTT\) algebras. Coincidences with fundamental \(R\) matrices allow to relate the Algebraic Bethe Ansatz for spinor and vector monodromy matrices.Gaussian unitary ensembles with two jump discontinuities, PDEs, and the coupled Painlevé II and IV systemshttps://zbmath.org/1472.820022021-11-25T18:46:10.358925Z"Lyu, Shulin"https://zbmath.org/authors/?q=ai:lyu.shulin"Chen, Yang"https://zbmath.org/authors/?q=ai:chen.yang.1Summary: We consider the Hankel determinant generated by the Gaussian weight with two jump discontinuities. Utilizing the results of \textit{C. Min} and \textit{Y. Chen} [Math. Methods Appl. Sci. 42, No. 1, 301--321 (2019; Zbl 1409.33018)] where a second-order partial differential equation (PDE) was deduced for the log derivative of the Hankel determinant by using the ladder operators adapted to orthogonal polynomials, we derive the coupled Painlevé IV system which was established in [\textit{X.-B. Wu} and \textit{S.-X. Xu}, Nonlinearity 34, No. 4, 2070--2115 (2021; Zbl 1470.34238)] by a study of the Riemann-Hilbert problem for orthogonal polynomials. Under double scaling, we show that, as \(n \rightarrow \infty\), the log derivative of the Hankel determinant in the scaled variables tends to the Hamiltonian of a coupled Painlevé II system and it satisfies a second-order PDE. In addition, we obtain the asymptotics for the recurrence coefficients of orthogonal polynomials, which are connected with the solutions of the coupled Painlevé II system.Approximate 1-norm minimization and minimum-rank structured sparsity for various generalized inverses via local searchhttps://zbmath.org/1472.901022021-11-25T18:46:10.358925Z"Xu, Luze"https://zbmath.org/authors/?q=ai:xu.luze"Fampa, Marcia"https://zbmath.org/authors/?q=ai:fampa.marcia-helena-c"Lee, Jon"https://zbmath.org/authors/?q=ai:lee.jon"Ponte, Gabriel"https://zbmath.org/authors/?q=ai:ponte.gabrielRecurrent neural network approach based on the integral representation of the Drazin inversehttps://zbmath.org/1472.920422021-11-25T18:46:10.358925Z"Stanimirović, Predrag S."https://zbmath.org/authors/?q=ai:stanimirovic.predrag-s"Živković, Ivan S."https://zbmath.org/authors/?q=ai:zivkovic.ivan-s"Wei, Yimin"https://zbmath.org/authors/?q=ai:wei.yiminSummary: In this letter, we present the dynamical equation and corresponding artificial recurrent neural network for computing the Drazin inverse for arbitrary square real matrix, without any restriction on its eigenvalues. Conditions that ensure the stability of the defined recurrent neural network as well as its convergence toward the Drazin inverse are considered. Several illustrative examples present the results of computer simulations.On the index of convergence of a class of Boolean matrices with structural propertieshttps://zbmath.org/1472.930822021-11-25T18:46:10.358925Z"Ramos, Guilherme"https://zbmath.org/authors/?q=ai:ramos.guilherme"Pequito, Sérgio"https://zbmath.org/authors/?q=ai:pequito.sergio"Caleiro, Carlos"https://zbmath.org/authors/?q=ai:caleiro.carlosSummary: Boolean matrices are of prime importance in the study of discrete event systems (DES), which allow us to model systems across a variety of applications. The index of convergence (i.e. the number of distinct powers of a Boolean matrix) is a crucial characteristic in that it assesses the transient behaviour of the system until reaching a periodic course. In this paper, adopting a graph-theoretic approach, we present bounds for the index of convergence of Boolean matrices for a diverse class of systems, with a certain decomposition. The presented bounds are an extension of the bound on irreducible Boolean matrices, and we provide non-trivial bounds that were unknown for classes of systems. Furthermore, the proposed method is able to determine the bounds in polynomial time. Lastly, we illustrate how the new bounds compare with the previously known bounds and we show their effectiveness in cases such as the benchmark IEEE 5-bus power system.Uniform recovery in infinite-dimensional compressed sensing and applications to structured binary samplinghttps://zbmath.org/1472.940202021-11-25T18:46:10.358925Z"Adcock, Ben"https://zbmath.org/authors/?q=ai:adcock.ben"Antun, Vegard"https://zbmath.org/authors/?q=ai:antun.vegard"Hansen, Anders C."https://zbmath.org/authors/?q=ai:hansen.anders-cSummary: Infinite-dimensional compressed sensing deals with the recovery of analog signals (functions) from linear measurements, often in the form of integral transforms such as the Fourier transform. This framework is well-suited to many real-world inverse problems, which are typically modeled in infinite-dimensional spaces, and where the application of finite-dimensional approaches can lead to noticeable artefacts. Another typical feature of such problems is that the signals are not only sparse in some dictionary, but possess a so-called local sparsity in levels structure. Consequently, the sampling scheme should be designed so as to exploit this additional structure. In this paper, we introduce a series of uniform recovery guarantees for infinite-dimensional compressed sensing based on sparsity in levels and so-called multilevel random subsampling. By using a weighted \(\ell^1\)-regularizer we derive measurement conditions that are sharp up to log factors, in the sense that they agree with the best known measurement conditions for oracle estimators in which the support is known a priori. These guarantees also apply in finite dimensions, and improve existing results for unweighted \(\ell^1\)-regularization. To illustrate our results, we consider the problem of binary sampling with the Walsh transform using orthogonal wavelets. Binary sampling is an important mechanism for certain imaging modalities. Through carefully estimating the local coherence between the Walsh and wavelet bases, we derive the first known recovery guarantees for this problem.