Recent zbMATH articles in MSC 15https://zbmath.org/atom/cc/152021-01-08T12:24:00+00:00WerkzeugA note on the Laplacian resolvent energy, Kirchhoff index and their relations.https://zbmath.org/1449.051832021-01-08T12:24:00+00:00"Zogić, Emir"https://zbmath.org/authors/?q=ai:zogic.emir"Glogić, Edin"https://zbmath.org/authors/?q=ai:glogic.edinSummary: Let \(G\) be a simple graph of order \(n\) and let \(L\) be its Laplacian matrix. Eigenvalues of the matrix \(L\) are denoted by \(\mu_1, \mu_2,\dots, \mu_n\) and it is assumed that \(\mu_1\ge \mu_2\ge \dots\ge \mu_n\). The Laplacian resolvent energy and Kirchhoff index of the graph \(G\) are defined as \(\mathrm{RL}(G) = \sum^n_{i=1} \frac{1}{n+1-\mu_i}\) and \(\mathrm{Kf}(G) = n \sum_{n-1}^{n-1}\frac{1}{\mu_i}\), respectively. In this paper, we derive some bounds on the invariant \(\mathrm{RL}(G)\) and establish a relation between \(\mathrm{RL}(G)\) and \(\mathrm{Kf}(G)\).Reciprocal matrices: properties and approximation by a transitive matrix.https://zbmath.org/1449.150812021-01-08T12:24:00+00:00"Bebiano, Natália"https://zbmath.org/authors/?q=ai:bebiano.natalia"Fernandes, Rosário"https://zbmath.org/authors/?q=ai:fernandes.rosario"Furtado, Susana"https://zbmath.org/authors/?q=ai:furtado.susanaSummary: Reciprocal matrices and, in particular, transitive matrices, appear in several applied areas. Among other applications, they have an important role in decision theory in the context of the analytical hierarchical process, introduced by \textit{T. L. Saaty} [J. Math. Psychol. 15, 234--281 (1977; Zbl 0372.62084)]. In this paper, we study the possible ranks of a reciprocal matrix and give a procedure to construct a reciprocal matrix with the rank and the off-diagonal entries of an arbitrary row (column) prescribed. We apply some techniques from graph theory to the study of transitive matrices, namely to determine the maximum number of equal entries, and distinct from \(\pm 1\), in a transitive matrix. We then focus on the \(n\)-by-\(n\) reciprocal matrix, denoted by \(C(n, x)\), with all entries above the main diagonal equal to \(x>0\). We show that there is a Toeplitz transitive matrix and a transitive matrix preserving the maximum possible number of entries of \(C(n, x)\), whose distances to \(C(n, x)\), measured in the Frobenius norm, are smaller than the one of the transitive matrix proposed by Saaty, constructed from the right Perron eigenvector of \(C(n, x)\). We illustrate our results with some numerical examples.A generalized convex quadratic programming to solve fuzzy linear system.https://zbmath.org/1449.902772021-01-08T12:24:00+00:00"Solaymani, Fard Omid"https://zbmath.org/authors/?q=ai:solaymani.fard-omid"Akhoundi, Naser"https://zbmath.org/authors/?q=ai:akhoundi.naser"Ramezanzadeh, Mohadeseh"https://zbmath.org/authors/?q=ai:ramezanzadeh.mohadeseh"Gachpazan, Morteza"https://zbmath.org/authors/?q=ai:gachpazan.mortezaSummary: The linear systems are one of the most important tools for modeling real-world phenomena. Because the real-world phenomena are always associated with uncertainty, solving the fuzzy linear system have a great importance. One of the proposed methods to find the exact and approximate solutions of a fuzzy linear system is using the least squares method. In this method, by choosing an arbitrary meter and solving a quadratic programming, they provide an approximate (or exact) solution for the fuzzy linear system. In this paper, at first, we prove that under some conditions and not depending on the selected meter the quadratic programming is convex. Therefore, by considering three different meters and solving several examples, we compare the obtained approximate solutions.Inferior limit of minimum eigenvalue of Fan product of M-matrices.https://zbmath.org/1449.150522021-01-08T12:24:00+00:00"Zhong, Qin"https://zbmath.org/authors/?q=ai:zhong.qinSummary: Fan product of matrices is one of the important problems in matrix theory. An estimator of inferior limit of minimum eigenvalue of Fan product of two nonsingular M-matrices is given by means of characteristic value containing domain theorem, and the result obtained depends only on the entries in the two nonsingular M-matrices, so that the computation is easy. Numerical example shows that the new estimator improved several existing results under certain conditions.A class of tensor eigenvalue complementarity problem.https://zbmath.org/1449.150142021-01-08T12:24:00+00:00"Luo, Gang"https://zbmath.org/authors/?q=ai:luo.gang"Yang, Qingzhi"https://zbmath.org/authors/?q=ai:yang.qingzhiSummary: In this paper, we generalize the matrix eigenvalue complementarity problem which has wide application in mechanical systems. A positive semidefinite eigenvalue complementarity problem (SDPEiCP) is established using fourth-order tensor form. Some properties, like the existence of the solution, computational complexity, are studied. We show the relation between SDPEiCP and a nonlinear constrained optimization problem. A shifted power method is proposed to compute the solution of SDPEiCP at last.Reverse-order law for core inverse of tensors.https://zbmath.org/1449.150602021-01-08T12:24:00+00:00"Sahoo, Jajati Keshari"https://zbmath.org/authors/?q=ai:sahoo.jajati-keshari"Behera, Ratikanta"https://zbmath.org/authors/?q=ai:behera.ratikantaSummary: The notion of the core inverse of tensors with the Einstein product was introduced, very recently [\textit{J. K. Sahoo} et al., ibid. 39, No. 1, Paper No. 9, 28 p. (2020; Zbl 1449.15061)]. In this paper we establish a few sufficient and necessary conditions for reverse-order law of this inverse. Further, we discuss mixed-type reverse-order law for core inverse. In addition to these, we discuss core inverse solutions of multilinear systems via the Einstein product. The prowess of the inverse is demonstrated to solve the Poisson problem in the multilinear system framework.On the positive definite solution of a class of pair of nonlinear matrix equations.https://zbmath.org/1449.150382021-01-08T12:24:00+00:00"Ali, Hasem"https://zbmath.org/authors/?q=ai:ali.hasem"Hossein, Sk M."https://zbmath.org/authors/?q=ai:hossein.sk-monowarSummary: We find some necessary and sufficient conditions for the existence of Hermitian positive definite solution of a pair of nonlinear matrix equations of the form:
\[\begin{aligned} X^{s_1}+A^*X^{-t_1}A+B^*Y^{-p_1}B=Q_1 \\ Y^{s_2}+A^*Y^{-t_2}A+B^*X^{-p_2}B=Q_2, \end{aligned}\]
and provide some algorithms for finding solutions. Finally, we give some numerical examples and study the convergence history of the iterations.Some inequalities on minimum eigenvalue for the Fan product of M-matrices.https://zbmath.org/1449.150502021-01-08T12:24:00+00:00"Huang, Kuanna"https://zbmath.org/authors/?q=ai:huang.kuanna"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.4|liu.hui.2|liu.hui.1|liu.hui.3"Han, Zhongming"https://zbmath.org/authors/?q=ai:han.zhongmingSummary: New lower bounds on the minimum eigenvalue for the Fan product of two M-matrices \(A\) and \(B\) are given. The estimating formulas of the bounds only depend on the entries of the M-matrices. Therefore, they are easy to calculate. Numerical examples show that the new formulas can improve some existing ones in some cases.On pair of generalized derivations in rings.https://zbmath.org/1449.160762021-01-08T12:24:00+00:00"Ali, Asma"https://zbmath.org/authors/?q=ai:ali.asma"Rahaman, Md. Hamidur"https://zbmath.org/authors/?q=ai:rahaman.md-hamidurSummary: Let \(R\) be an associative ring with extended centroid \(C\), let \(G\) and \(F\) be generalized derivations of \(R\) associated with nonzero derivations \(\delta\) and \(d\), respectively, and let \(m, k, n \geq1\) be fixed integers. In the present paper, we study the situations: (i) \(F(x)\circ_mG(y)=(x \circ_n y)^k\), (ii) \([F(x),y]_m+[x,d(y)]_n=0\) for all \(y\), \(x\) in some appropriate subset of \(R\).Jordan standard form of generalized cubic matrix.https://zbmath.org/1449.150322021-01-08T12:24:00+00:00"Lv, Hongbin"https://zbmath.org/authors/?q=ai:lv.hongbin"Yang, Zhongpeng"https://zbmath.org/authors/?q=ai:yang.zhongpeng"Chen, Meixiang"https://zbmath.org/authors/?q=ai:chen.meixiang"Feng, Xiaoxia"https://zbmath.org/authors/?q=ai:feng.xiaoxiaSummary: By using the method of matrix analysis, based on the existing defined expressions of the generalized cubic matrix, we gave the equivalent expressions related to all possible eigenvalues of the generalized cubic matrices, and gave the complete description of Jordan standard form of the generalized cubic matrix.Bounds for the largest singular value of nonnegative rectangular tensors.https://zbmath.org/1449.150222021-01-08T12:24:00+00:00"Sang, Caili"https://zbmath.org/authors/?q=ai:sang.caili"Zhao, Jianxing"https://zbmath.org/authors/?q=ai:zhao.jianxingSummary: For estimates of the largest singular value \({\lambda_0} (\mathcal{A})\) of a nonnegative rectangular tensor \(\mathcal{A}\), using the arbitrariness of some selected elements of \(\mathcal{A}\), classification discussion idea and some techniques of inequalities, new bounds for \({\lambda_0} (\mathcal{A})\) are given and proved to be an improvement of some existing results. The obtained results are verified by numerical examples, which show that the obtained bounds are more accurate than some existing results.The property and application of real representation of parabolic commutative quaternion matrices.https://zbmath.org/1449.150462021-01-08T12:24:00+00:00"Kong, Xiangqiang"https://zbmath.org/authors/?q=ai:kong.xiangqiangSummary: In this paper, on the basis of real representation of parabolic commutative quaternion, the real representation of parabolic commutative matrices is given. By using the matrix representation, the sufficient and necessary conditions for the existence of eigenvalues of a commutative quaternion matrix and the Gerschgorin disk theorem are obtained. Furthermore, several calculation properties of the commutative quaternion matrix are obtained. Finally, the validity of the conclusion is verified by an example.Generalized public transportation scheduling using max-plus algebra.https://zbmath.org/1449.150662021-01-08T12:24:00+00:00"Subiono"https://zbmath.org/authors/?q=ai:subiono.s"Fahim, Kistosil"https://zbmath.org/authors/?q=ai:fahim.kistosil"Adzkiya, Dieky"https://zbmath.org/authors/?q=ai:adzkiya.diekySummary: In this paper, we discuss the scheduling of a wide class of transportation systems. In particular, we derive an algorithm to generate a regular schedule by using max-plus algebra. Inputs of this algorithm are a graph representing the road network of public transportation systems and the number of public vehicles in each route. The graph has to be strongly connected, which means there is a path from any vertex to every vertex. Let us remark that the algorithm is general in the sense that we can allocate any number of vehicles in each route. The algorithm itself consists of two main steps. In the first step, we use a novel procedure to construct the model. Then in the second step, we compute a regular schedule by using the power algorithm. We describe our proposed framework for an example.A non-monotone adaptive trust region method for generalized eigenvalues of symmetric tensors.https://zbmath.org/1449.650842021-01-08T12:24:00+00:00"Yang, Yueting"https://zbmath.org/authors/?q=ai:yang.yueting"Wang, Li"https://zbmath.org/authors/?q=ai:wang.li.6|wang.li.5|wang.li|wang.li.1|wang.li.4|wang.li.3|wang.li.2"Xing, Fu'na"https://zbmath.org/authors/?q=ai:xing.funa"Chen, Yuting"https://zbmath.org/authors/?q=ai:chen.yuting"Cao, Mingyuan"https://zbmath.org/authors/?q=ai:cao.mingyuanSummary: The generalized eigenvalue of symmetric tensors is transformed into a homogeneous polynomial optimization problem on unit hyper-sphere. Using projection idea and combining with adaptive technology, an adaptive trust region method is proposed for finding the largest (smallest) generalized eigenvalue of symmetric tensors. Global convergence of the proposed algorithm and second-order necessary conditions of the optimal solutions are established, respectively. Numerical experiments are listed to illustrate the efficiency of the proposed method. When the generalized eigenvalue problem degenerates to Z-eigenvalue problem, the numerical comparison with the existing results shows that the algorithm is more effective.Strong anti-transitive fuzzy matrix and its properties.https://zbmath.org/1449.150722021-01-08T12:24:00+00:00"Gu, Yundong"https://zbmath.org/authors/?q=ai:gu.yundong"Wang, Bin"https://zbmath.org/authors/?q=ai:wang.bin.2|wang.bin.1|wang.bin.4|wang.bin|wang.bin.3"Zhao, Feng"https://zbmath.org/authors/?q=ai:zhao.fengSummary: Firstly, the significance of anti-transitive matrix in network analysis is discussed. On this basis, the definition of strong anti-transitive matrix is proposed. Then, its equivalent characterizations, graphic characteristic, cut-matrix properties, standard form period and index are studied. Finally, an algorithm for finding the strong anti-transitive circle is proposed.Pseudo-spectra localization sets of tensors.https://zbmath.org/1449.150132021-01-08T12:24:00+00:00"He, Jun"https://zbmath.org/authors/?q=ai:he.jun"Liu, Yanmin"https://zbmath.org/authors/?q=ai:liu.yanminSummary: Pseudo-spectrum of tensors plays an important role in the stability of homogeneous dynamical system. The properties of the pseudo-spectrum of tensors are further studied by the definition of pseudo-spectrum of tensors. New localization sets for the pseudo-spectrum of tensors are given. It is shown that the new sets are tighter than the existing ones.Properties of a class of perturbed Toeplitz periodic tridiagonal matrices.https://zbmath.org/1449.150692021-01-08T12:24:00+00:00"Fu, Yaru"https://zbmath.org/authors/?q=ai:fu.yaru"Jiang, Xiaoyu"https://zbmath.org/authors/?q=ai:jiang.xiaoyu"Jiang, Zhaolin"https://zbmath.org/authors/?q=ai:jiang.zhaolin"Jhang, Seongtae"https://zbmath.org/authors/?q=ai:jhang.seongtaeSummary: In this paper, for a class of perturbed Toeplitz periodic tridiagonal (PTPT) matrices, some properties, including the determinant, the inverse matrix, the eigenvalues and the eigenvectors, are studied in detail. Specifically, the determinant of the PTPT matrix can be explicitly expressed using the well-known Fibonacci numbers; the inverse of the PTPT matrix can also be explicitly expressed using the Lucas number and only four elements in the PTPT matrix. Eigenvalues and eigenvectors can be obtained under certain conditions. In addition, some algorithms are presented based on these theoretical results. Comparison of our new algorithms and some recent works is given. Numerical results confirm our new theoretical results and show that the new algorithms not only can obtain accurate results but also have much better computing efficiency than some existing algorithms studied recently.\(E\)-eigenvalue localization sets for tensors.https://zbmath.org/1449.150172021-01-08T12:24:00+00:00"Sang, Caili"https://zbmath.org/authors/?q=ai:sang.caili"Chen, Zhen"https://zbmath.org/authors/?q=ai:chen.zhenSummary: Several existing \(Z\)-eigenvalue localization sets for tensors are first generalized to \(E\)-eigenvalue localization sets. And then two tighter \(E\)-eigenvalue localization sets for tensors are presented. As applications, a sufficient condition for the positive definiteness of fourth-order real symmetric tensors, a sufficient condition for the positive semi-definiteness of fourth-order real symmetric tensors, and a new upper bound for the \(Z\)-spectral radius of weakly symmetric nonnegative tensors are obtained. Finally, numerical examples are given to verify the theoretical results.\(Z\)-eigenvalue exclusion theorems for tensors.https://zbmath.org/1449.150252021-01-08T12:24:00+00:00"Wang, Gang"https://zbmath.org/authors/?q=ai:wang.gang.5|wang.gang|wang.gang.4|wang.gang.1|wang.gang.2|wang.gang.3"Zhang, Yuan"https://zbmath.org/authors/?q=ai:zhang.yuanSummary: To locate all \(Z\)-eigenvalues of a tensor more precisely, we establish three \(Z\)-eigenvalue exclusion sets such that all \(Z\)-eigenvalues do not belong to them and get three tighter \(Z\)-eigenvalue inclusion sets of tensor by using these \(Z\)-eigenvalue exclusion sets. Furthermore, we show that the new inclusion sets are tighter than the existing results via two running examples.\(Z\)-eigenvalue inclusion theorem of tensors and the geometric measure of entanglement of multipartite pure states.https://zbmath.org/1449.150262021-01-08T12:24:00+00:00"Xiong, Liang"https://zbmath.org/authors/?q=ai:xiong.liang"Liu, Jianzhou"https://zbmath.org/authors/?q=ai:liu.jianzhouSummary: In our paper, we concentrate on the \(Z\)-eigenvalue inclusion theorem and its application in the geometric measure of entanglement of multipartite pure states. We present a new \(Z\)-eigenvalue inclusion theorem by virtue of the division and classification of tensor elements, and tighter bounds of \(Z\)-spectral radius of weakly symmetric nonnegative tensors are obtained. As applications, we present some theoretical upper and lower bounds of entanglement for symmetric pure state with nonnegative amplitudes for two kinds of geometric measures with different definitions, respectively.Solving fuzzy linear systems by a block representation of generalized inverse: the core inverse.https://zbmath.org/1449.150732021-01-08T12:24:00+00:00"Jiang, Hongjie"https://zbmath.org/authors/?q=ai:jiang.hongjie"Wang, Hongxing"https://zbmath.org/authors/?q=ai:wang.hongxing"Liu, Xiaoji"https://zbmath.org/authors/?q=ai:liu.xiaojiSummary: This paper presents a method for solving fuzzy linear systems, where the coefficient matrix is an \(n\times n\) real matrix, using a block structure of the Core inverse, and we use the Hartwig-Spindelböck decomposition to obtain the Core inverse of the coefficient matrix \(A\). The aim of this paper is twofold. First, we obtain a strong fuzzy solution of fuzzy linear systems, and a necessary and sufficient condition for the existence strong fuzzy solution of fuzzy linear systems are derived using the Core inverse of the coefficient matrix \(A\). Second, general strong fuzzy solutions of fuzzy linear systems are derived, and an algorithm for obtaining general strong fuzzy solutions of fuzzy linear systems by Core inverse is also established. Finally, some examples are given to illustrate the validity of the proposed method.\(\boldsymbol{X}\)-simplicity of interval max-min matrices.https://zbmath.org/1449.150632021-01-08T12:24:00+00:00"Berežný, Štefan"https://zbmath.org/authors/?q=ai:berezny.stefan"Plavka, Ján"https://zbmath.org/authors/?q=ai:plavka.janSummary: A matrix \(A\) is said to have \(\boldsymbol{X}\)-simple image eigenspace if any eigenvector \(x\) belonging to the interval \(\boldsymbol{X}=\{x:\underline{x}\leq x\leq\overline{x}\}\) containing a constant vector is the unique solution of the system \(A\otimes y=x\) in \(\boldsymbol{X}\). The main result of this paper is an extension of \(\boldsymbol{X}\)-simplicity to interval max-min matrix \(\boldsymbol{A}=\{A:\underline{A}\leq A\leq\overline{A}\}\) distinguishing two possibilities, that at least one matrix or all matrices from a given interval have \(\boldsymbol{X}\)-simple image eigenspace. \(\boldsymbol{X}\)-simplicity of interval matrices in max-min algebra are studied and equivalent conditions for interval matrices which have \(\boldsymbol{X}\)-simple image eigenspace are presented. The characterized property is related to and motivated by the general development of tropical linear algebra and interval analysis, as well as the notions of simple image set and weak robustness (or weak stability) that have been studied in max-min and max-plus algebras.A note on resolving the inconsistency of one-sided max-plus linear equations.https://zbmath.org/1449.150652021-01-08T12:24:00+00:00"Li, Pingke"https://zbmath.org/authors/?q=ai:li.pingkeThe author studies one-sided linear systems of equations over max-plus algebra. For an inconsistent system, he proposes to modify, in a minimal way, the right-hand side vector to achieve solvability. The studied problem is formulated as a mixed integer linear program and illustrated by several small examples.
Reviewer: Katarína Cechlárová (Košice)RS rings and their applications.https://zbmath.org/1449.160712021-01-08T12:24:00+00:00"Wu, Cang"https://zbmath.org/authors/?q=ai:wu.cang"Zhao, Liang"https://zbmath.org/authors/?q=ai:zhao.liang.1By an RS ring, the authors mean an associative ring \(R\) with unity, in which every regular element is strongly regular. The main results of this paper are the following: (i) The class of RS rings properly contains the class of nilpotent-closed rings. (ii) Necessary and sufficient conditions for IC rings and von Neumann regular rings to be RS rings are given. (iii) \(R\) is abelian if, and only if, \(R\) is RS and the set of all regular elements of \(R\) is multiplicatively closed. (iv) \(R\) is reduced if, and only if, \(R\) is RS and every \(\pi\)-regular element is regular in \(R\).
Reviewer: Tibor Juhász (Eger)Matrix representations of Sturm-Liouville problems with distribution potentials on time scales.https://zbmath.org/1449.343212021-01-08T12:24:00+00:00"Liu, Nana"https://zbmath.org/authors/?q=ai:liu.nana"Ao, Jijun"https://zbmath.org/authors/?q=ai:ao.jijunSummary: The matrix representations of second order Sturm-Liouville problems with distribution potentials on bounded time scales are investigated. The corresponding equivalences between Sturm-Liouville problems with distribution potentials on time scales and a certain kind of matrix eigenvalue problems are obtained. Both of the separated and coupled self-adjoint boundary conditions are considered to obtain the main results.An \(S\)-type singular value inclusion set for rectangular tensors.https://zbmath.org/1449.150212021-01-08T12:24:00+00:00"Sang, Caili"https://zbmath.org/authors/?q=ai:sang.caili"Zhao, Jianxing"https://zbmath.org/authors/?q=ai:zhao.jianxingSummary: In this paper, we study the locations of all singular values of rectangular tensors. By breaking the index set of rectangular tensors into disjoint nonempty proper subsets \(S\) and its complement, and using classification discussion idea and some techniques of inequalities, we obtain an \(S\)-type singular value inclusion set for rectangular tensors, which is an improvement of some existing results. Finally, numerical examples are given to verify the theoretical results and show that the new set is more accurate than some existing sets.A Bauer-Hausdorff matrix inequality.https://zbmath.org/1449.150552021-01-08T12:24:00+00:00"Yan, Zizong"https://zbmath.org/authors/?q=ai:yan.zizong"Wu, Shanhe"https://zbmath.org/authors/?q=ai:wu.shanheSummary: We present a biorthogonal process for two subspaces of \(\mathbb{C}^n\). Applying this process, we derive a matrix inequality, which generalizes the Bauer-Hausdorff inequality for vectors and includes the Wang-IP inequality for matrices. Meanwhile, we obtain its equivalent matrix inequality.Modified Hermitian-normal splitting iteration methods for a class of complex symmetric linear systems.https://zbmath.org/1449.650552021-01-08T12:24:00+00:00"Du, Ya-Kun"https://zbmath.org/authors/?q=ai:du.ya-kun"Qin, Mei"https://zbmath.org/authors/?q=ai:qin.meiSummary: In this paper, the modifications of the Hermitian-Normal splitting iteration methods for solving a class of complex symmetric linear systems are presented. Theoretical analysis shows that the modified iteration methods of Hermitian-normal splitting are unconditionally convergent; the coefficient matrices of the two linear systems solved in each iteration of the methods are real symmetric positive definite. Inexact version of the methods employs the Krylov subspace method as an internal iteration to accelerate. Numerical examples from two model problems are given to illustrate the effectiveness of the modified iteration methods.Yet another generalization of Sylvester's theorem and its application.https://zbmath.org/1449.110932021-01-08T12:24:00+00:00"Laishram, Shanta"https://zbmath.org/authors/?q=ai:laishram.shanta"Ngairangbam, Sudhir Singh"https://zbmath.org/authors/?q=ai:ngairangbam.sudhir-singh"Singh, Maibam Ranjit"https://zbmath.org/authors/?q=ai:singh.maibam-ranjitSummary: In this paper, we consider Sylvester's theorem on the largest prime divisor of a product of consecutive terms of an arithmetic progression, and prove another generalization of this theorem. As an application of this generalization, we provide an explicit method to find perfect powers in a product of terms of binary recurrence sequences and associated Lucas sequences whose indices come from consecutive terms of an arithmetic progression. In particular, we prove explicit results for Fibonacci, Jacobsthal, Mersenne and associated Lucas sequences.Inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their optimal approximations.https://zbmath.org/1449.650892021-01-08T12:24:00+00:00"Xu, Wei-Ru"https://zbmath.org/authors/?q=ai:xu.weiru"Chen, Guo-Liang"https://zbmath.org/authors/?q=ai:chen.guoliangSummary: In this paper, the inverse eigenvalue problems for skew-Hermitian reflexive and anti-reflexive matrices and their associated optimal approximation problems which are constrained by their partially prescribed eigenpairs are considered, respectively. First, the necessary and sufficient conditions of the solvability for the inverse eigenvalue problems of skew-Hermitian reflexive and anti-reflexive matrices are both derived, and the general solutions are also presented. Then the solutions of the corresponding optimal approximation problems in the Frobenius norm to a given matrix are also given, respectively. Furthermore, we give the algorithms to compute the optimal approximate skew-Hermitian reflexive and anti-reflexive solutions and present some illustrative numerical examples.Inertia formulae related to the generalized inverse \(A_{T,S}^{(2)}\) with applications.https://zbmath.org/1449.150082021-01-08T12:24:00+00:00"Wu, Zhongcheng"https://zbmath.org/authors/?q=ai:wu.zhongcheng"He, Ningxin"https://zbmath.org/authors/?q=ai:he.ningxinSummary: In this paper, firstly, we establish the inertia formulae for some matrix expressions related to the generalized inverse \(A_{T, S}^{(2)}\). Then, as applications, based on the derived inertia formulae, we study the definiteness of some matrices. Finally, necessary and sufficient conditions for some matrices to be positive definite, positive semidefinite, negative definite and negative semi-definite are given, respectively.Idempotent basis and involutory basis of \(n \times n\) matrix space.https://zbmath.org/1449.150342021-01-08T12:24:00+00:00"Su, Ruyan"https://zbmath.org/authors/?q=ai:su.ruyan"Yang, Zhongpeng"https://zbmath.org/authors/?q=ai:yang.zhongpeng"Chen, Meixiang"https://zbmath.org/authors/?q=ai:chen.meixiangSummary: When \(n \ge 2\), there exists not only a basis of \(n \times n\) matrix space composed of idempotent matrices, but also infinitely many different idempotent bases. This leads to the conclusion that each \(n \times n\) matrix can be uniquely linearly represented by idempotent matrices, and the linear expression coefficients are shown. This paper also gives the definition of base rank of basis of \(n \times n\) matrix, and proves that the maximum lower bound and the minimum upper bound of the base rank of all idempotent bases are reachable. The involution base of \(n \times n\) matrix space is obtained firstly.The tensor splitting methods for solving tensor absolute value equation.https://zbmath.org/1449.150592021-01-08T12:24:00+00:00"Bu, Fan"https://zbmath.org/authors/?q=ai:bu.fan"Ma, Chang-Feng"https://zbmath.org/authors/?q=ai:ma.changfengSummary: Recently, \textit{W. Li} et al. [Appl. Numer. Math. 134, 105--121 (2018; Zbl 1432.65037)] presented the tensor splitting methods for solving multilinear systems and \textit{S. Du} et al. [Sci. China, Math. 61, No. 9, 1695--1710 (2018; Zbl 1401.15024)] generalized tensor absolute value equations. In this paper, we verify the existence of solutions of tensor absolute value equations and propose the tensor splitting methods for solving this class of equation. Furthermore, the convergence analysis of the tensor splitting method is also studied under suitable conditions. Finally, numerical examples show that our algorithm is an efficient iterative method.A new iteration method for solving non-Hermitian positive definite linear systems.https://zbmath.org/1449.650622021-01-08T12:24:00+00:00"Nasabzadeh, Hamideh"https://zbmath.org/authors/?q=ai:nasabzadeh.hamidehSummary: In this paper, based on the single-step Hermitian and Skew-Hermitian (SHSS) iteration method [\textit{C.-X. Li} and \textit{S.-L. Wu}, Appl. Math. Lett. 44, 26--29 (2015; Zbl 1315.65032)] and by using the generalized Taylor expansion method for solving linear systems [\textit{F. Toutounian} and the author, Appl. Math. Comput. 248, 602--609 (2014; Zbl 1338.65087)], a new method (GT-SHSS) is introduced to solve non-Hermitian positive definite linear systems. The convergence properties of the new method are discussed. We show that by using suitable parameters, the GT-SHSS iteration method is faster than the corresponding SHSS iteration method. The numerical examples confirm the effectiveness of the new method.\(E\)-eigenvalue inclusion sets for tensors and their applications.https://zbmath.org/1449.150202021-01-08T12:24:00+00:00"Sang, Caili"https://zbmath.org/authors/?q=ai:sang.caili"Zhao, Jianxing"https://zbmath.org/authors/?q=ai:zhao.jianxingSummary: For locations of \(E\)-eigenvalues of tensors, some \(E\)-eigenvalue inclusion sets for tensors are obtained by using techniques of inequalities. The result generalizes and improves some known existing results. As an application, a more accurate upper bound for the \(Z\)-spectral radius of weakly symmetric nonnegative tensors is obtained.A modified alternately linearized implicit iteration method for M-matrix algebraic Riccati equation.https://zbmath.org/1449.650962021-01-08T12:24:00+00:00"Guan, Jinrui"https://zbmath.org/authors/?q=ai:guan.jinrui"Zhou, Fang"https://zbmath.org/authors/?q=ai:zhou.fang"Zubair, Ahmed"https://zbmath.org/authors/?q=ai:zubair.ahmedSummary: In this paper, we study the numerical solution of M-matrix algebraic Riccati equation (MARE). Based on the alternately linearized implicit iteration method, we propose a modified alternately linearized implicit iteration method (MALI) for computing the minimal nonnegative solution of MARE. The convergence of the MALI iteration method is proved under suitable conditions. The convergence rate with optimal parameters is given for the MARE associated with a nonsingular M-matrix or an irreducible singular M-matrix. Numerical experiments are given to show that the MALI iteration method is feasible in some cases.Updated preconditioned Hermitian and skew-Hermitian splitting-type iteration methods for solving saddle-point problems.https://zbmath.org/1449.650512021-01-08T12:24:00+00:00"Chen, Fang"https://zbmath.org/authors/?q=ai:chen.fang"Li, Tian-Yi"https://zbmath.org/authors/?q=ai:li.tianyi"Lu, Kang-Ya"https://zbmath.org/authors/?q=ai:lu.kang-yaSummary: For the saddle-point problems, we discuss and analyze the updated preconditioned Hermitian and skew-Hermitian splitting (UPHSS) iteration method in detail. On this basis, we introduce a two-stage iteration method for the UPHSS iteration method. Theoretical analysis shows that the UPHSS and two-stage UPHSS iteration methods are convergent to the unique solution of the saddle-point linear system when the parameter is suitably chosen. Numerical examples show the correctness of the theory and the effectiveness of these methods.A new estimation for eigenvalues of matrix power functions.https://zbmath.org/1449.150512021-01-08T12:24:00+00:00"Kian, M."https://zbmath.org/authors/?q=ai:kian.mohsen"Bakherad, M."https://zbmath.org/authors/?q=ai:bakherad.mojtabaSummary: We give an estimation for the eigenvalues of matrix power functions. In particular, it has been shown that \[\lambda((A+B)^p)\leq \lambda(2^{p-1}(A^p+B^p-\gamma I))\;\; (p\geq 2)\] for all positive semi-definite matrices \(A, B\), where \(\gamma\) is a positive constant. This provides a sharper bound for the known estimation for eigenvalues.An improvement on ``New practical criteria for nonsingular \(H\)-matrices''.https://zbmath.org/1449.150802021-01-08T12:24:00+00:00"Tuo, Qing"https://zbmath.org/authors/?q=ai:tuo.qingSummary: In this paper, by adopting the new positive diagonal factor, several new sufficient conditions for nonsingular \(H\)-matrices are obtained. Meanwhile, these results also improve and generalize the main achievements of [\textit{Q. Tuo} et al., Acta Math. Appl. Sin. 31, No. 1, 143--151 (2008; Zbl 1174.15015)]. Numerical examples are presented to illustrate the efficiency of our results.Dual self-conjugate solution of the quaternion Lyapunov equation \(AX + X{A^*} = B\).https://zbmath.org/1449.150412021-01-08T12:24:00+00:00"Huang, Jingpin"https://zbmath.org/authors/?q=ai:huang.jingpin"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min.2|wang.min.1|wang.min"Wang, Yun"https://zbmath.org/authors/?q=ai:wang.yunSummary: This paper discusses dual self-conjugate solution of the quaternion Lyapunov equation \(AX + X{A^*} = B\). The original problem is transformed into an equation with self-conjugate structure by using structural properties of dual self-conjugate matrix and matrix transformations. Necessary and sufficient conditions for the existence of a dual self-conjugate solution and the general solution of the equation are obtained by the vectorization of the self-conjugate matrix. The results expand solution forms of Lyapunov equation, and the numerical examples demonstrate the effectiveness of the proposed algorithm.Properties of hyperbola-type commutative quaternion matrix and finding approach of its inverse matrix.https://zbmath.org/1449.150752021-01-08T12:24:00+00:00"Kong, Xiangqiang"https://zbmath.org/authors/?q=ai:kong.xiangqiang"Jiang, Tongsong"https://zbmath.org/authors/?q=ai:jiang.tongsongSummary: Taking the concept of hyperbola-type commutative quaternion and its matrix as a basis, the serial properties of the hyperbola-type commutative quaternion and its real representation are found. Then the serial properties of the hyperbola-type commutative quaternion matrix are derived. By introducing the real representation of the matrix, the finding method of inverse matrix is obtained for the hyperbolic commutative quaternion matrix. And the correctness of this method is verified with a numerical computation examples.Extremal ranks of complex components in general solutions of the matrix equation \(AXB = C\) over quaternion field.https://zbmath.org/1449.150422021-01-08T12:24:00+00:00"Lian, Dezhong"https://zbmath.org/authors/?q=ai:lian.dezhong"Xie, Jinshan"https://zbmath.org/authors/?q=ai:xie.jinshanSummary: By using a complex representation of quaternion matrix \(\Phi (\cdot)\), the linear matrix equation \(AXB = C\) over the quaternion field is changed into the matrix equation \(\Phi (A)\widetilde{X}\Phi (B) = \Phi (C)\) over the complex field. Then according to general solutions of this complex matrix equation and numerous properties regarding extreme ranks of block matrix, formulas of extreme ranks of complex matrices \(\{X_0\}\), \(\{X_1\}\) are established. These complex matrices are complex components of general solutions \(X = X_0 + X_1 j\) of the quaternion matrix equation. As an application, we give necessary and sufficient conditions for following special cases: there exists at least a complex matrix \(\widetilde{X}\) in general solutions of the matrix equation; and all general solutions of the matrix equation are complex ones.Matrix power means and Pólya-Szegő type inequalities.https://zbmath.org/1449.150532021-01-08T12:24:00+00:00"Kian, Mohsen"https://zbmath.org/authors/?q=ai:kian.mohsen"Rashid, Fatemeh"https://zbmath.org/authors/?q=ai:rashid.fatemehSummary: It is shown that, if \(\mu\) is a compactly supported probability measure on \(\mathbb{M}^+_n\), then, for every unit vector \(\eta\in\mathbb{C}^n\), there exists at compactly supported probability measure (denoted by \(\langle\mu\eta,\eta\rangle)\) on \(\mathbb{R}^+\) so that the inequality \[\langle P_t(\mu)\eta,\eta\rangle\le P_t(\langle\mu \eta,\eta\rangle),\ t\in(0,1],\] holds. In particular, we consider a reverse of the above inequality and present some Pólya-Szegő type inequalities for power means of probability measures on positive matrices.Matrix method of calculating the rescattering wave in a periodic structure.https://zbmath.org/1449.780052021-01-08T12:24:00+00:00"Belyaev, Yuriĭ Nikolaevich"https://zbmath.org/authors/?q=ai:belyaev.yurii-nikolaevichSummary: A method of recursive calculation of the characteristic matrix of the layered structure is developed. Analytical solutions for the matrix of second, third and fourth order describing the propagation of waves in periodic structures are found.Maximum eigenvalue algorithm of nonnegative matrix under diagonal similarity transformation.https://zbmath.org/1449.650832021-01-08T12:24:00+00:00"Wang, Xincun"https://zbmath.org/authors/?q=ai:wang.xincun"Lv, Hongbin"https://zbmath.org/authors/?q=ai:lv.hongbin"Shang, Yuying"https://zbmath.org/authors/?q=ai:shang.yuyingSummary: By introducing a parameter, we constructed a positive diagonal matrix related to the row sum of the iteration matrix. By using the positive diagonal similarity transformation of matrices, we gave a numerical algorithm for computing the maximum eigenvalue and corresponding eigenvectors of irreducible nonnegative matrices. The selection of parameters in each step of the algorithm is flexible and the convergence speed is improved.Schur factorization and orthogonal diagonal factorization of quasi-row (column) symmetric matrices.https://zbmath.org/1449.150372021-01-08T12:24:00+00:00"Yuan, Huiping"https://zbmath.org/authors/?q=ai:yuan.huipingSummary: The author considered the Schur factorization, orthogonal diagonal factorization, Hermite matrix factorization and generalized inverse of quasi-row (column)symmetric matrices, gave the formulas of the Schur factorization, orthogonal diagonal factorization, Hermite matrix factorization and generalized inverse of quasi-row (column) symmetric matrices. The calculation results show that the method not only reduces the amount of calculation and storage, but also does not reduce the numerical accuracy.The DQMR algorithm for Drazin inverse solution of singular linear systems.https://zbmath.org/1449.650912021-01-08T12:24:00+00:00"Deng, Yong"https://zbmath.org/authors/?q=ai:deng.yongSummary: In recent years, the algorithm of Drazin inverse solution for singular linear systems has caused wide concern by many scholars, and many research results that depend on the Krylov subspace are obtained. However, the Krylov subspace method is very cumbersome and it is very difficult to solve the singular linear incompatible system. Based on this, in this paper, the DQMR algorithm for Drazin inverse solution of compatible or non-compatible singular linear systems \(Ax=b\) is given by using projection method, where \(A\in \mathbb{C}^{n\times n}\) is a singular Hermitian matrix with arbitrary index. The DQMR algorithm of singular systems is similar to the QMR algorithm of non-singular systems.Core and core-EP inverses of tensors.https://zbmath.org/1449.150612021-01-08T12:24:00+00:00"Sahoo, Jajati Keshari"https://zbmath.org/authors/?q=ai:sahoo.jajati-keshari"Behera, Ratikanta"https://zbmath.org/authors/?q=ai:behera.ratikanta"Stanimirović, Predrag S."https://zbmath.org/authors/?q=ai:stanimirovic.predrag-s"Katsikis, Vasilios N."https://zbmath.org/authors/?q=ai:katsikis.vasilios-n"Ma, Haifeng"https://zbmath.org/authors/?q=ai:ma.haifengSummary: Specific definitions of the core and core-EP inverses of complex tensors are introduced. Some characterizations, representations and properties of the core and core-EP inverses are investigated. The results are verified using specific algebraic approach, based on proposed definitions and previously verified properties. The approach used here is new even in the matrix case.An improved Collatz-Wielandt algorithm for computing maximum eigenvalue of nonnegative matrix.https://zbmath.org/1449.650812021-01-08T12:24:00+00:00"Shang, Yuying"https://zbmath.org/authors/?q=ai:shang.yuying"Zhang, Meili"https://zbmath.org/authors/?q=ai:zhang.mei-li"Lv, Hongbin"https://zbmath.org/authors/?q=ai:lv.hongbin"Wang, Xincun"https://zbmath.org/authors/?q=ai:wang.xincunSummary: By using the property of irreducible nonnegative matrices and Collatz-Wielandt function, an improved C-W algorithm for computing the maximum eigenvalue of irreducible nonnegative matrices is presented. The algorithm has a good convergence rate when the parameters are properly selected.Two-criteria problems for optimal protection of elastic structures from vibration.https://zbmath.org/1449.903092021-01-08T12:24:00+00:00"Balandin, D. V."https://zbmath.org/authors/?q=ai:balandin.dmitry-v"Ezhov, E. N."https://zbmath.org/authors/?q=ai:ezhov.e-n"Fedotov, I. A."https://zbmath.org/authors/?q=ai:fedotov.i-aThe problem of vibroprotection of a sky-scrapper is tackled as the problem of two-criteria optimization. The first criterion characterizes displacement of the ground floor with respect to building's basement, and the second describes deformation of a many-mass system. The mathematical model of the building's vibration includes a system of ODEs with uncontrollable and controllable actions with feedback in its right-hand side. To obtain Pareto-optimal damper, the apparatus of matrix linear inequalities is used. A new criterion of optimality is formed out of two initial criteria by means of the Germeyer convolution.
Reviewer: Aleksey Syromyasov (Saransk)Lower bound for the minimum eigenvalue of nonsingular \(M\)-matrix.https://zbmath.org/1449.150312021-01-08T12:24:00+00:00"Zhong, Qin"https://zbmath.org/authors/?q=ai:zhong.qinSummary: Estimation of the bounds of the minimum eigenvalue of nonsingular \(M\)-matrix is important part in the theory of matrix. A new lower bound of the minimum eigenvalue of nonsingular \(M\)-matrix was given by using Hölder inequality. The new estimating formula was easy to calculate since it only depended on the entries of \(M\)-matrix. Numerical results showed that the new lower bound estimates could improve considerably some previous results.Dirac-Yang-Mills model equations with a spinor gauge symmetry.https://zbmath.org/1449.810322021-01-08T12:24:00+00:00"Marchuk, Nikolaĭ Gur'evich"https://zbmath.org/authors/?q=ai:marchuk.nikolaj-gurevichSummary: In the developed model where spin \(1/2\) fermions acquire masses by an interaction with (spin 1) gauge field with spinor symmetry. Particle mass is determined by the constant interaction of the particle with the gauge field.Solutions to a class of equations in integral matrices of order 2.https://zbmath.org/1449.150442021-01-08T12:24:00+00:00"Yin, Qianqian"https://zbmath.org/authors/?q=ai:yin.qianqian"Liang, Xinran"https://zbmath.org/authors/?q=ai:liang.xinran"Yuan, Pingzhi"https://zbmath.org/authors/?q=ai:yuan.pingzhiSummary: To solve the problems of integer matrix equations related to Pythagorean equation \({x^2} + {y^2} = {z^2}\), the solutions \( (X, Y)\) to the \(2 \times 2\) integral matrix equation \(X^2 + Y^2 = \lambda I\), where \(\lambda \in \mathbb{Z}\) and \(I\) is the unit matrix, which are related to the Pythagorean equation, are investigated and completely solved by using the basic operation of matrix to transform the problem of integer matrix equation into the problem of solving some Diophantine equations, which is gradually extended from the special case to the general case. The solutions to \(2 \times 2\) integral matrix equation \(X^2 - Y^2 = \lambda I\) also can be solved with similar methods.The properties of determinants for matrix multiplications over commutative semirings.https://zbmath.org/1449.150772021-01-08T12:24:00+00:00"Liu, Yijin"https://zbmath.org/authors/?q=ai:liu.yijin"Wang, Xueping"https://zbmath.org/authors/?q=ai:wang.xueping.1Summary: This paper mainly investigates the properties of determinants for matrix multiplications over commutative semirings. It discusses the relationships between the determinant of matrix multiplications and the multiplication of determinants for matrices, and shows the relationships between the multiplication of adjoint matrices and the adjoint matrix of matrix multiplications.A new preconditioner for solving weighted Toeplitz least squares problems.https://zbmath.org/1449.650432021-01-08T12:24:00+00:00"Cheng, Guo"https://zbmath.org/authors/?q=ai:cheng.guo"Li, Jicheng"https://zbmath.org/authors/?q=ai:li.jichengSummary: In this paper, we study a fast algorithm for solving the weighted Toeplitz least squares problems. Firstly, on the basis of the augmented linear system, we develop a new SIMPLE-like preconditioner for solving such linear systems. Secondly, the convergence of the iterative method is studied, and used to prove that all eigenvalues of the preconditioned matrix are real and nonunit eigenvalues are located in a positive interval. Again, we also study the eigenvector distribution and the degree of the minimal polynomial of the preconditioned matrix. Finally, related numerical experiments are carried out to show that the new preconditioner is more effective than some existing preconditioners.The relationship between similar invariant subspaces and invariant subspaces.https://zbmath.org/1449.470222021-01-08T12:24:00+00:00"Gu, Wen"https://zbmath.org/authors/?q=ai:gu.wen"Ni, Junna"https://zbmath.org/authors/?q=ai:ni.junnaSummary: The concept of ``similar invariant subspace'' is defined and the relationship between similar invariant subspace and invariant subspace under the conditions of reversible linear transformation and general linear transformation is discussed. Using the theory of vector space, it is proved that similar invariant subspace is equivalent to invariant subspace under the condition of reversible linear transformation. Furthermore, it is proved that for a linear transformation \(\sigma\) of vector space \(V\), if \(W\) is a similar invariant subspace, then \(W\) must be an invariant subspace.Hybrid enriched bidiagonalization for discrete ill-posed problems.https://zbmath.org/1449.650932021-01-08T12:24:00+00:00"Hansen, Per Christian"https://zbmath.org/authors/?q=ai:hansen.per-christian"Dong, Yiqiu"https://zbmath.org/authors/?q=ai:dong.yiqiu"Abe, Kuniyoshi"https://zbmath.org/authors/?q=ai:abe.kuniyoshiSummary: The regularizing properties of the Golub-Kahan bidiagonalization algorithm are powerful when the associated Krylov subspace captures the dominating components of the solution. In some applications the regularized solution can be further improved by enrichment, that is, by augmenting the Krylov subspace with a low-dimensional subspace that represents specific prior information. Inspired by earlier work on GMRES, we demonstrate how to carry these ideas over to the bidiagonalization algorithm, and we describe how to incorporate Tikhonov regularization. This leads to a hybrid iterative method where the choice of regularization parameter in each iteration also provides a stopping rule.Non-additive Lie centralizer of infinite strictly upper triangular matrices.https://zbmath.org/1449.160542021-01-08T12:24:00+00:00"Aiat Hadj, D. A."https://zbmath.org/authors/?q=ai:aiat-hadj.driss-ahmedSummary: Let \(\mathcal{F}\) be an field of zero characteristic and \(N_\infty(\mathcal{F})\) be the algebra of infinite strictly upper triangular matrices with entries in \(\mathcal{F}\), and \(f:N_\infty(\mathcal{F})\rightarrow N_\infty(\mathcal{F})\) be a non-additive Lie centralizer of \(N_\infty(\mathcal{F})\), that is, a map satisfying that \(f([X,Y])=[f(X),Y]\) for all \(X,Y\in N_\infty(\mathcal{F})\). We prove that \(f(X)=\lambda X\), where \(\lambda \in \mathcal{F}\).A note on the forward order law for least square \(g\)-inverse of three matrix products.https://zbmath.org/1449.150042021-01-08T12:24:00+00:00"Liu, Zhongshan"https://zbmath.org/authors/?q=ai:liu.zhongshan"Xiong, Zhiping"https://zbmath.org/authors/?q=ai:xiong.zhiping"Qin, Yingying"https://zbmath.org/authors/?q=ai:qin.yingyingSummary: In this paper, by using the expressions for maximal ranks of the generalized Schur complement, we obtain some necessary and sufficient conditions for the forward order laws \(A_1\{1,3\}A_2\{1,3\}A_3\{1,3\}\subseteq (A_1A_2A_3)\{1,3\}\) and \(A_1\{1,4\}A_2\{1,4\}A_3\{1,4\}\subseteq (A_1A_2A_3)\{1,4\}\).Exponents of the primitive Boolean matrices with fixed girth.https://zbmath.org/1449.051782021-01-08T12:24:00+00:00"Yu, Guanglong"https://zbmath.org/authors/?q=ai:yu.guanglong"Guo, Shuguang"https://zbmath.org/authors/?q=ai:guo.shuguang"Jia, Wenjuan"https://zbmath.org/authors/?q=ai:jia.wenjuan"Che, Yuhan"https://zbmath.org/authors/?q=ai:che.yuhanSummary: The girth of a primitive Boolean matrix is defined to be the girth of its associated digraph. In this paper, among all primitive Boolean matrices of order \(n\), the primitive exponents of those of girth \(g\) are considered. For the primitive matrices of both order \(n\geq 10\) and girth \(g>\frac{n^2-4n}{4(n-3)}\), the matrices with primitive exponents in \([2n-2+(g-1)(n-3),n+g(n-2)]\) are completely characterized.Theorem on the norm of elements of spinor groups.https://zbmath.org/1449.150582021-01-08T12:24:00+00:00"Shirokov, Dmitriĭ Sergeevich"https://zbmath.org/authors/?q=ai:shirokov.dmitry-sSummary: In this article we consider Clifford's algebra over the field of real numbers of finite dimension. We define the operation of Hermitian conjugation for the elements of Clifford's algebra. This operation allows us to define the structure of Euclidian space on the Clifford algebra. We consider pseudo-orthogonal group and its subgroups -- special pseudo-orthogonal, orthochronous, orthochorous and special orthochronous groups. As known, spinor groups are double covers of these orthogonal groups. We proved a theorem that relates the norm of element of spinor group with the minor of matrix of the orthogonal group.Introducing \(p\)-eigenvectors; exact solutions for some simple matrices.https://zbmath.org/1449.150562021-01-08T12:24:00+00:00"Lócsi, Levente"https://zbmath.org/authors/?q=ai:locsi.leventeSummary: A common way to define a norm of a matrix is to take the supremum of the fraction of the vector norms of the matrix-vector product and the nonzero vector, with respect to a given vector norm, i.e. the least upper bound for the norm of the vectors of the transformed unit sphere. In this paper we examine the above mentioned fraction, defining induction curves and surfaces, we show that there exist some vectors, such that this fraction is independent of the applied \(p\)-norm (and are not eigenvectors). These are to be called \(p\)-eigenvectors. Exact solutions are constructed for some simple matrices. No previous work was found in this topic so far.Belitskii's reduction and standard form of matrix pairs.https://zbmath.org/1449.150352021-01-08T12:24:00+00:00"Zhang, Chao"https://zbmath.org/authors/?q=ai:zhang.chao.1|zhang.chao.9|zhang.chao.4|zhang.chao.7|zhang.chao.2|zhang.chao.3|zhang.chao.5|zhang.chao.8|zhang.chao.6|zhang.chao"Cai, Hongyan"https://zbmath.org/authors/?q=ai:cai.hongyanSummary: In the representation theory, the algebra \(\Gamma = k\langle {x, y} \rangle\) plays an important role in the research of the representation type of algebras. In this paper, all the representations of \(\Gamma\) up to isomorphisms are described by using the Belitskii's reduction. Equivalently, we determine the standard form of matrix pairs of size three. As an application, we obtain the number of parameters of this linear matrix problem based on the standard form.The matrix depict approaches of rough sets.https://zbmath.org/1449.030312021-01-08T12:24:00+00:00"Liu, Wenjun"https://zbmath.org/authors/?q=ai:liu.wenjun-j"Guo, Qing"https://zbmath.org/authors/?q=ai:guo.qingSummary: In this paper, the authors study the rough sets using matrix. Firstly, the upper and lower approximation operators of Pawlak rough sets, fuzzy-rough sets, rough-fuzzy sets are redefined using the matrix representation. Then the properties of these upper and lower approximation operators are discussed. At last, the definitions of the upper and lower approximation operators of Pawlak rough sets, fuzzy-rough sets, rough-fuzzy sets are unified through the compound operator of matrix. The matrix approach affords not only a simple computing method of the upper and lower approximation operators of rough sets, but also a new inference method.Note on the relations of the positive/negative determinants \(|I + XY|\) over commutative semirings.https://zbmath.org/1449.150112021-01-08T12:24:00+00:00"Liu, Yijin"https://zbmath.org/authors/?q=ai:liu.yijin"Wu, Li"https://zbmath.org/authors/?q=ai:wu.li"Wang, Xueping"https://zbmath.org/authors/?q=ai:wang.xueping.1Summary: In this paper, the relations of the positive/negative determinants \(|I + XY|\) over commutative semirings are investigated.Eigenpairs of a family of tridiagonal matrices: three decades later.https://zbmath.org/1449.150122021-01-08T12:24:00+00:00"Da Fonseca, C. M."https://zbmath.org/authors/?q=ai:da-fonseca.carlos-martins"Kowalenko, V."https://zbmath.org/authors/?q=ai:kowalenko.victorThis is a brief survey, where the authors review some important applications of the eigenpairs formulas for a family of tridiagonal matrices based on the paper of \textit{L. Losonczi} [Acta Math. Hung. 60, No. 3--4, 309--322 (1992; Zbl 0771.15004)] and show that several recent problems have been solved with reference to the paper of \textit{W.-C. Yueh} [Appl. Math. E-Notes 5, 66--74 (2005; Zbl 1068.15006)], which was published after Losonczi's paper [loc. cit.].
Reviewer: Mohammad Sal Moslehian (Mashhad)\(S\)-type inclusion set of generalized eigenvalue for tensors.https://zbmath.org/1449.150182021-01-08T12:24:00+00:00"Sang, Caili"https://zbmath.org/authors/?q=ai:sang.caili"Zhao, Jianxing"https://zbmath.org/authors/?q=ai:zhao.jianxingSummary: Let \(\{A, B\}\) be an \(m\)th-order \(n\)-dimensional regular tensor pair. By breaking \(N = \{1, 2, \cdots, n\}\) into disjoint subsets \(S\) and its complement \(\bar S = N/S\), and using classification discussion idea, the arbitrariness of the selection of some elements of tensor pair \(\{A, B\}\) and scaling techniques of inequalities, the location for the eigenvalues of tensor pair \(\{A, B\}\) was solved, and an \(S\)-type inclusion set of eigenvalue for tensor pair \(\{A, B\}\) was given. Numerical results show that the obtained inclusion set is more accurate than existing inclusion sets.The reverse order law for the generalized Bott-Duffin inverses of matrices.https://zbmath.org/1449.150022021-01-08T12:24:00+00:00"Cao, Zhiqiang"https://zbmath.org/authors/?q=ai:cao.zhiqiang"Du, Nailin"https://zbmath.org/authors/?q=ai:du.nailinSummary: In this paper we investigate the reverse order law for the generalized Bott-Duffin inverses of two L-p.s.d. matrices. The concept of L-p.s.d. matrices is a generalization of semi-positive definiteness, in the sense that \({\mathbb{R}}^n\)-p.s.d. matrices are exactly semi-positive definite matrices. We establish a relationship between generalized Bott-Duffin inverses and \(A_{S, {S^\bot}}^{ (2)}\) inverses. By using their properties, we give necessary and sufficient conditions for the reverse order law of the generalized Bott-Duffin inverses. In the end we give two explicit examples to test our results.Further discussion and application on matrix which is similar to its power.https://zbmath.org/1449.150292021-01-08T12:24:00+00:00"Yan, Yumin"https://zbmath.org/authors/?q=ai:yan.yumin"Yang, Zhongpeng"https://zbmath.org/authors/?q=ai:yang.zhongpeng"Chen, Meixiang"https://zbmath.org/authors/?q=ai:chen.meixiang"Lv, Hongbin"https://zbmath.org/authors/?q=ai:lv.hongbin"Yu, Zhizheng"https://zbmath.org/authors/?q=ai:yu.zhizhengSummary: For any nonzero integer \(k\), a complex matrix \(A\) is similar to its power \({A^k}\) if its all eigenvalues are 1. Furthermore, the necessary and sufficient conditions under which the complex matrix \(A\) is similar to its power \({A^k}\) (\(k \in \mathbb{Z}^+\)) are given. The existence of the solution to a class of nonlinear matrix equations is proved by applications.Some new results for Chebyshev matrix polynomials of first kind.https://zbmath.org/1449.330132021-01-08T12:24:00+00:00"Shehata, Ayman"https://zbmath.org/authors/?q=ai:shehata.aymanSummary: The main aim of the present paper is to investigate some new relations and generating matrix functions for Chebyshev matrix polynomials of the first kind. Some consequences of our main results are also discussed.Third Hankel determinant for a class of generalized analytic functions related with lemniscate of Bernoulli and symmetric points.https://zbmath.org/1449.300352021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Ma, Li'na"https://zbmath.org/authors/?q=ai:ma.linaSummary: In this paper, we introduce a class of generalized analytic functions, denoted by \(SL_s^* (\alpha, \mu)\), which are associated with lemniscate of Bernoulli and symmetric points. We investigate the Hankel determinant \({H_3} (1)\) for these functions and the upper bound of the above determinant is obtained.Bounds on the structural indices of primitive non-powerful generalized sign pattern matrices.https://zbmath.org/1449.150782021-01-08T12:24:00+00:00"Huang, Yufei"https://zbmath.org/authors/?q=ai:huang.yufeiSummary: In view of the special effects of ``loop'' in the study of structural index problems, two classes of special generalized signed digraphs are defined: primitive non-powerful generalized signed digraphs with intersecting cycles structure and that with distinguished intersecting cycles structure, respectively. With restriction on primitive nonpowerful generalized signed digraphs with intersecting cycles structure and those with distinguished intersecting cycles structure, upper bounds on the structural indices, e.g. \(k\)th local \(\tau\)-base, \(k\)th same \(\tau\)-base, \(k\)th lower \(\tau\)-base, \(k\)th upper \(\tau\)-base and \(\omega\)-indecomposable base, are discussed by imitating the digraphs, analyzing the ambiguous reachable set and using the properties of Frobenius numbers, respectively.Idempotent and nilpotent matrices of triangular modules on the distributive lattice.https://zbmath.org/1449.150092021-01-08T12:24:00+00:00"Li, Aimei"https://zbmath.org/authors/?q=ai:li.aimei"Wu, Miaoling"https://zbmath.org/authors/?q=ai:wu.miaoling"Wang, Yaxian"https://zbmath.org/authors/?q=ai:wang.yaxianSummary: In this paper, we introduce some vital properties which include irreflexivity, reflexivity and idempotence of \(T\)-idempotent matrix and \(S\)-idempotent matrix over the distributive lattice. In addition, we also give the sufficient conclusions under which irreflexive matrix becomes \(T\)-nilpotent matrix and \(S\)-nilpotent matrix, and prove them with our own method and improved method.A new class of positive semi-definite tensors.https://zbmath.org/1449.150282021-01-08T12:24:00+00:00"Xu, Yi"https://zbmath.org/authors/?q=ai:xu.yi"Liu, Jinjie"https://zbmath.org/authors/?q=ai:liu.jinjie"Qi, Liqun"https://zbmath.org/authors/?q=ai:qi.liqunSummary: In this paper, a new class of positive semi-definite tensors, the MO tensor, is introduced. It is inspired by the structure of Moler matrix, a class of test matrices. Then we focus on two special cases in the MO-tensors: Sup-MO tensor and essential MO tensor. They are proved to be positive definite tensors. Especially, the smallest H-eigenvalue of a Sup-MO tensor is positive and tends to zero as the dimension tends to infinity, and an essential MO tensor is also a completely positive tensor.The least-square solutions to the linear matrix equations \(AX = B\), \(YA = D\) with \( (R, S_\sigma)\)-commutative matrices.https://zbmath.org/1449.150432021-01-08T12:24:00+00:00"Wen, Yaqiong"https://zbmath.org/authors/?q=ai:wen.yaqiong"Li, Jiaofen"https://zbmath.org/authors/?q=ai:li.jiaofen"Li, Wen"https://zbmath.org/authors/?q=ai:li.wen.1Summary: The definition of the \( (R, S_\sigma)\)-commutative matrix was given by a previous author in 2012. This paper discusses the general structure of \( (R, S_\sigma)\)-commutative matrix, and the least squares problem and the best approximation problem of linear equations \(AX = B\), \(YA = D\) in the set of \( (R, S_\sigma)\)-commutative matrices for the given matrices \(X, Y, B, D\). Then we analyze the expressions of the least-square commutative solution and the best approximate solution of \( (R, S_\sigma)\)-commutative matrix in detail. At the same time, the necessary and sufficient conditions for the existence of the \( (R, S_\sigma)\)-commutative solution are analyzed when the linear matrix equations are consistent.Relative position of three subspaces in a Hilbert space.https://zbmath.org/1449.460232021-01-08T12:24:00+00:00"Enomoto, Masatoshi"https://zbmath.org/authors/?q=ai:enomoto.masatoshi"Watatani, Yasuo"https://zbmath.org/authors/?q=ai:watatani.yasuoSummary: We study the relative position of three subspaces in an infinite dimensional Hilbert space. In the finite-dimensional case over an arbitrary field, \textit{S. Brenner} [J. Algebra 6, 100--114 (1967; Zbl 0229.16020)] described the general position of three subspaces completely. We extend it to a certain class of three subspaces in an infinite-dimensional Hilbert space over the complex numbers.Two groups of mixed reverse order laws for generalized inverses of two and three matrix products.https://zbmath.org/1449.150052021-01-08T12:24:00+00:00"Tian, Yongge"https://zbmath.org/authors/?q=ai:tian.yonggeSummary: Generalized inverses of a matrix product can be written as certain matrix expressions that are composed by the given matrices and their generalized inverses, and a challenging task in this respect is to establish various reasonable reverse order laws for generalized inverses of matrix products. In this paper, we present two groups of known and new mixed reverse order laws for the Moore-Penrose inverses of products of two and three matrices through various conventional matrix operations. We also establish four groups of matrix set inclusions that are composed by \(\{1\}\)- and \(\{1,2\}\)-generalized inverses of \(A\), \(B\), \(C\), and their products \(AB\) and \(ABC\).A continued fractional recurrence algorithm for generalized inverse tensor Padé approximation.https://zbmath.org/1449.410112021-01-08T12:24:00+00:00"Gu, Chuanqing"https://zbmath.org/authors/?q=ai:gu.chuanqing"Huang, Yizheng"https://zbmath.org/authors/?q=ai:huang.yizheng"Chen, Zhibing"https://zbmath.org/authors/?q=ai:chen.zhibingSummary: The tensor exponential function has been widely used in cybernetics, image processing and various engineering fields. Based on the generalized matrix inverse, an effective tensor generalized inverse is defined for the first time on the scalar inner product space, and a continued fractional algorithm is constructed for the tensor Padé approximation. On the other hand, we successfully use the tensor \(t\)-product to calculate the power of the tensor, and recursively give the power series expansion of the tensor exponential function. Based on the previous two work, the continuous fractional algorithm designed in this paper is used to approximate the tensor exponential function. Its characteristic is that the algorithm can be programmed to implement recursive calculations, and in the calculation process, it is not necessary to calculate the product of the tensor and to calculate the inverse of the tensor. The numerical experiments of the two tensor exponential functions given in this paper show that comparing the continuous fractional algorithm with the commonly used truncation method, the proposed algorithm is effective without reducing the approximation order. If the dimension of the tensor is relatively large, a continuous fractional algorithm based on the generalized inverse of tensors will also have certain advantages.Third-order Hankel determinant for analytic functions based on the left-half of lemniscate of Bernoulli.https://zbmath.org/1449.300362021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huo"Niu, Xiaomeng"https://zbmath.org/authors/?q=ai:niu.xiaomengSummary: In this paper, we introduce a class of analytic functions \(B{L_\alpha} (0 \le \alpha \le 1)\) in the left-half bounded domain of lemniscate of Bernoulli, which are defined by using subordination relationship. We then investigate the third Hankel determinant \(\boldsymbol{H}_3 (1)\) of \(B{L_\alpha}\), and obtain its upper bound. In addition, some special cases of the results are given.Application of the alternating direction method for the structure-preserving finite element model updating problem.https://zbmath.org/1449.650902021-01-08T12:24:00+00:00"Zhang, Baocan"https://zbmath.org/authors/?q=ai:zhang.baocanSummary: This paper shows that the alternating direction method can be used to solve the structured inverse quadratic eigenvalue problem with symmetry, positive semi-definiteness and sparsity requirements. The results of numerical examples show that the proposed method works well.Beginnings of matrix theory in the Czech Republic and their results.https://zbmath.org/1449.150012021-01-08T12:24:00+00:00"Štěpánová, Martina"https://zbmath.org/authors/?q=ai:stepanova.martinaThe book traces the early history of matrix theory in the Czech lands approximately in the period 1850--1950. The main emphasis is on the outstanding results of Eduard Weyr (1852--1903) dealing with the so-called Weyr characteristic and Weyr canonical form. These concepts are closely related to the well-known Segre characteristic and Jordan canonical form of a square matrix. Despite Weyr's priority over Jordan, his results remained almost forgotten for about a century. The Weyr characteristic has become increasingly popular since the 1980s thanks to H. Schneider, D. Hershkowitz, and their followers, who were studying relations between matrices and their associated graphs. The Weyr canonical form has been rediscovered several times, and became more widely known due to \textit{H. Shapiro}'s paper [Am. Math. Mon. 106, No. 10, 919--929 (1999; Zbl 0981.15008)], who introduced it to nonspecialists and gave proper credit to Weyr.
The text represents an expanded version of the author's dissertation thesis. It is based on a careful study of a large number of primary sources spanning the period of about 150 years. Weyr's results are first described in a proper historical context, and then explained once again in the language of modern mathematics. A considerable portion of the book is devoted to the reception of Weyr's work since its publication until the present day. The book is very well written and will be of interest to all historians of mathematics. Readers who are not fluent in Czech can take advantage of a twenty-page-long English summary.
Reviewer: Antonín Slavík (Praha)Nonsurjective maps between rectangular matrix spaces preserving disjointness, triple products, or norms.https://zbmath.org/1449.150672021-01-08T12:24:00+00:00"Li, Chi-Kwong"https://zbmath.org/authors/?q=ai:li.chi-kwong"Tsai, Mimg-Cheng"https://zbmath.org/authors/?q=ai:tsai.mimg-cheng"Wang, Ya-Shu"https://zbmath.org/authors/?q=ai:wang.ya-shu"Wong, Ngai-Ching"https://zbmath.org/authors/?q=ai:wong.ngai-ching|wong.ngaichingThis paper offers solutions to some new linear preserver problems (for some historical background, see [the first author and \textit{N.-K. Tsing}, Linear Algebra Appl. 162--164, 217--235 (1992; Zbl 0762.15016); the first author and \textit{S. Pierce}, Am. Math. Mon. 108, No. 7, 591--605 (2001; Zbl 0991.15001)]). Let \(M_{m,n}\) be the \(\mathbb{F}\)-space of all \(m\times n\) matrices over \(\mathbb{F}\) where \(\mathbb{F=R}\) or \(\mathbb{C}\). We say that \(A,B\in M_{m,n}\) are disjoint (written \(A\bot B\)) if \(AB^{\ast}=0\) and \(BA^{\ast}=0\). An \(\mathbb{F}\)-linear mapping \(\varphi : M_{m,n}\rightarrow M_{r,s}\) is said to preserve disjointness if \(A\bot B\) \(\implies\varphi(A)\bot\varphi(B)\) for all \(A,B\in M_{m,n}\). The main theorem of this paper is that \(\varphi\) preserves disjointness if and only if there exist diagonal matrices \(Q_{1},Q_{2}\) with positive diagonal entries such that for some unitary matrices \(U\) and \(V\) of suitable sizes we have \[ \varphi(A)=U\left[ \begin{matrix} A\otimes Q_{1} & 0 & 0\\ 0 & A\otimes Q_{2} & 0\\ 0 & 0 & 0 \end{matrix} \right] V\text{ for all }A\in M_{m,n} \] (\(Q_{1}\) or \(Q_{2}\) may be vacuous). A linear mapping \(\varphi:M_{m,n} \rightarrow M_{r,s}\) is called a \(JB^{\ast}\)-triple homomorphism if \(\varphi(AB^{\ast}C+CB^{\ast}A)=\varphi(A)\varphi(B)^{\ast}\varphi (C)+\varphi(C)\varphi(B)^{\ast}\varphi(A)\) for all \(A,B,C\in M_{m,n}\). Using the main theorem, the authors show that there is a similar description for \(\varphi\) when \(\varphi\) is a \(JB^{\ast}\)-triple homomorphism (an extra hypothesis on the size of the zero block in the bottom right-hand corner of the displayed matrix is required). Further results are obtained for linear preservers \(\varphi\) for the Schatten \(p\)-norm and the Ky-Fan \(k\)-norm. However in the latter case, the result only holds for the case \(\mathbb{F} =\mathbb{C}\) since real isometries for Ky-Fan \(k\)-norms do not preserve disjointness.
Reviewer: John D. Dixon (Ottawa)Real representation and inverse matrix method of the mixed type commutative quaternion matrix.https://zbmath.org/1449.150032021-01-08T12:24:00+00:00"Kong, Xiangqiang"https://zbmath.org/authors/?q=ai:kong.xiangqiang"Jiang, Tongsong"https://zbmath.org/authors/?q=ai:jiang.tongsongSummary: Based on the concept of mixed type commutative quaternions, firstly, the real representation of the mixed type commutative quaternions is given. Secondly, the series properties of the real representation of the mixed type commutative matrix are derived, and the necessary and sufficient conditions for the existence of the eigenvalues are given. Finally, the inverse matrix of the mixed type commutative matrix is obtained, and the correctness of the conclusion is verified by an example.A generalized Hermitian nonnegative-definite solution on matrix equation \(AXB = C\).https://zbmath.org/1449.150392021-01-08T12:24:00+00:00"Deng, Yong"https://zbmath.org/authors/?q=ai:deng.yongSummary: In this paper, a sufficient and necessary condition for the existence of generalized Hermitian nonnegative definite solution of matrix equation \(AXB = C\) is given, and its expression is derived. This result is applied to two instances. Compared with the generalized Hermitian nonnegative definite solutions obtained by this paper and a literature, it is found that the solution given in the literature is wrong. Meanwhile, the general covariance structure of two matrices with quadratic random independence is given.Verified computation of eigenpairs in the generalized eigenvalue problem for nonsquare matrix pencils.https://zbmath.org/1449.650792021-01-08T12:24:00+00:00"Miyajima, Shinya"https://zbmath.org/authors/?q=ai:miyajima.shinyaSummary: This paper considers an optimization problem arising from the generalized eigenvalue problem \(Ax = \lambda Bx\), where \(A, B \in \mathbb{C}^{m \times n}\) and \(m > n\). Some previous authors showed that the optimization problem can be solved by utilizing right singular vectors of \(C:=[B,A]\). In this paper, we focus on computing intervals containing the solution. When some singular values of \(C\) are multiple or nearly multiple, we can enclose bases of corresponding invariant subspaces of \({C^H}C\), where \({C^H}\) denotes the conjugate transpose of \(C\), but cannot enclose the corresponding right singular vectors. The purpose of this paper is to prove that the solution can be obtained even when we utilize the bases instead of the right singular vectors. Based on the proved result, we propose an algorithm for computing the intervals. Numerical results show property of the algorithm.Credibility verification of \(Z\)-eigenpairs of symmetric tensors based on inverse-free Newton's method.https://zbmath.org/1449.150622021-01-08T12:24:00+00:00"Sang, Haifeng"https://zbmath.org/authors/?q=ai:sang.haifeng"Li, Min"https://zbmath.org/authors/?q=ai:li.min.10|li.min.8|li.min.9|li.min.1|li.min.6|li.min|li.min.3|li.min.2|li.min.4|li.min.7|li.min.5"Liu, Panpan"https://zbmath.org/authors/?q=ai:liu.panpan"Wang, Chunyan"https://zbmath.org/authors/?q=ai:wang.chunyan"Luan, Tian"https://zbmath.org/authors/?q=ai:luan.tianSummary: By using the inverse-free Newton's method and the interval algorithm theory, we studied the credibility verification of \(Z\)-eigenpairs of symmetric tensors, and proposed an interval algorithm to calculate \(Z\)-eigenpairs. The algorithm output an approximate \(Z\)-eigenpair and its corresponding error bound, so that an exact \(Z\)-eigenpair of symmetric tensors must exist within the error range of the approximate solution.On generalized Lie derivations.https://zbmath.org/1449.160772021-01-08T12:24:00+00:00"Bennis, Driss"https://zbmath.org/authors/?q=ai:bennis.driss"Vishki, Hamid Reza Ebrahimi"https://zbmath.org/authors/?q=ai:vishki.hamid-reza-ebrahimi"Fahid, Brahim"https://zbmath.org/authors/?q=ai:fahid.brahim"Bahmani, Mohammad Ali"https://zbmath.org/authors/?q=ai:bahmani.mohammad-aliSummary: In this paper, we investigate generalized Lie derivations. We give a complete characterization of when each generalized Lie derivation is a sum of a generalized inner derivation and a Lie derivation. This generalizes a result given by \textit{D. Benkovič} [Linear Algebra Appl. 434, No. 6, 1532--1544 (2011; Zbl 1216.16032)]. We also investigate when every generalized Lie derivation on some particular kind of unital algebras is a sum of a generalized derivation and a central map which vanishes on all commutators. Precisely, we consider both the unital algebras with nontrivial idempotents and the trivial extension algebras.Kernel of a class of linearized polynomials.https://zbmath.org/1449.111082021-01-08T12:24:00+00:00"Jin, Yong"https://zbmath.org/authors/?q=ai:jin.yong"Jiang, Jingjing"https://zbmath.org/authors/?q=ai:jiang.jingjingSummary: Let \(\boldsymbol{C}\) be the companion matrix of a linearized polynomial and \(\boldsymbol{I}\) be the identity matrix of the same order. We gave a necessary and sufficient condition for its reversibility by analyzing the structure of \(\boldsymbol{C} + \boldsymbol{I}\). Based on this, we gave linearized polynomials whose kernel had trivial intersection with the kernel of trace function.Upper bounds for \(||A^{-1}||_\infty\) of eventually Nekrasov matrices.https://zbmath.org/1449.150572021-01-08T12:24:00+00:00"She, Lili"https://zbmath.org/authors/?q=ai:she.lili"Zhao, Jianxing"https://zbmath.org/authors/?q=ai:zhao.jianxingSummary: A new class of nonsingular matrices, called eventually Nekrasov matrices, is defined. By using some existing upper bounds of the infinity norm of the inverse of Nekrasov matrices, upper bounds for \(||A^{-1}||_\infty\) of an eventually Nekrasov matrix \(A\) are given, and proved to be an generalization and improvement of some existing bounds. A numerical example is given to show that the bounds are also applicable to those classes that are not the class of nonsingular H-matrices, and that the bounds could reach the true value of \(||A^{-1}||_\infty\) in some cases.Pseudo-unitary matrices and the \(*\)-structures of Radford algebra.https://zbmath.org/1449.150472021-01-08T12:24:00+00:00"Li, Shiyu"https://zbmath.org/authors/?q=ai:li.shiyu"Zhou, Hainan"https://zbmath.org/authors/?q=ai:zhou.hainan"Shen, Wenjie"https://zbmath.org/authors/?q=ai:shen.wenjie"Chen, Huixiang"https://zbmath.org/authors/?q=ai:chen.huixiangSummary: The 8-dimensional Radford algebra over the complex number field is a Hopf algebra whose \(*\)-structures are determined by complex \(2\times 2\)-matrices \(A\) satisfying \(\bar AA = I\). Such matrices are called pseudo-unitary matrices. The two \(*\)-structures determined by two pseudo-unitary matrices are equivalent if and only if the two pseudo-unitary matrices satisfy an equivalence relation \(\sim\). In this paper, the pseudo-unitary \(2\times 2\)-matrices are studied and classified with respect to the equivalence relation \(\sim\). It is shown that any pseudo-unitary \(2\times 2\)-matrix is equivalent to the identity matrix with respect to \(\sim\). Consequently, up to the equivalence of \(*\)-structures, the 8-dimensional Radford algebra has a unique Hopf \(*\)-algebra structure.Asymptotic eigenvalue estimation for a class of structured matrices.https://zbmath.org/1449.150702021-01-08T12:24:00+00:00"Liang, Juan"https://zbmath.org/authors/?q=ai:liang.juan"Lai, Jiangzhou"https://zbmath.org/authors/?q=ai:lai.jiangzhou"Niu, Qiang"https://zbmath.org/authors/?q=ai:niu.qiangSummary: In this paper we consider eigenvalue asymptotic estimations for a class of structured matrices arising from statistical applications. The asymptotic upper bounds of the largest eigenvalue \( (\lambda_{\max})\) and the sum of squares of eigenvalues \( (\sum\limits_{i = 1}^n {\lambda_i^2})\) are derived. Both these bounds are useful in examining the stability of certain Markov process. Numerical examples are provided to illustrate tightness of the bounds.A new matrix inversion for Bell polynomials and its applications.https://zbmath.org/1449.150062021-01-08T12:24:00+00:00"Wang, Jin"https://zbmath.org/authors/?q=ai:wang.jinSummary: The present paper gives a new Bell matrix inversion which arises from the classical Lagrange inversion formula. Some new relations for the Bell polynomials are obtained, including a Bell matrix inversion in closed form and an inverse form of the classical Faa di Bruno formula.Toeplitz solution of Sylvester equation and its optimal approximation over quaternion field.https://zbmath.org/1449.150402021-01-08T12:24:00+00:00"Huang, Jingpin"https://zbmath.org/authors/?q=ai:huang.jingpin"Lan, Jiaxin"https://zbmath.org/authors/?q=ai:lan.jiaxin"Mao, Liying"https://zbmath.org/authors/?q=ai:mao.liying"Wang, Min"https://zbmath.org/authors/?q=ai:wang.min|wang.min.2|wang.min.1Summary: In this paper, we study the Toeplitz matrix solution of Sylvester equation and its optimal approximation over quaternion field. By using the real representation of a quaternion matrix and Kronecker product of matrices, the necessary and sufficient condition for the existence of a Toeplitz matrix solution and the general solution of the quaternion Sylvester equation \(AX - XB = C\) are obtained. Meanwhile, in the Toeplitz solution set, the expression of the optimal approximation solution to the given quaternion Toeplitz matrix is derived.Depiction technology of super corona distance matrix spectrum.https://zbmath.org/1449.150272021-01-08T12:24:00+00:00"Xu, Xiaojing"https://zbmath.org/authors/?q=ai:xu.xiaojing"Wang, Peiwen"https://zbmath.org/authors/?q=ai:wang.peiwen"Wang, Zhiping"https://zbmath.org/authors/?q=ai:wang.zhipingSummary: The topological structure of the graph has important significance in chemical molecular structure, and the various matrices of the graph contain the topology information of the graph. In this paper, we depict the distance spectra of four kinds of double corona according to the definition of distance spectrum and the structure characteristics of four types of graphs. The corresponding distance matrix is obtained by mathematical induction. On the basic of them, the block matrix is constructed, which is a super corona distance matrix. By using the uniqueness of the matrix eigenvalues, the eigenvalues and eigenvectors of the super corona distance matrix are solved, and the accuracy and reliability of the conclusion are verified simultaneously. Finally, we research the distance spectra of \(G^{ (S)} \circ \{{G_1}, {G_2}\}\), \(G^{ (Q)} \circ \{{G_1}, {G_2}\}\), \(G^{ (R)} \circ \{{G_1}, {G_2}\}\), \(G^{ (T)} \circ \{{G_1}, {G_2}\}\), when \(G\) is a complete graph and \({G_1}, {G_2}\) are regular graphs.New perturbation bounds for factor R of the SR factorization.https://zbmath.org/1449.150362021-01-08T12:24:00+00:00"Li, Ping"https://zbmath.org/authors/?q=ai:li.ping.1|li.ping.2|li.ping|li.ping.3|li.ping.5|li.ping.4"Yu, Xiaofei"https://zbmath.org/authors/?q=ai:yu.xiaofeiSummary: The SR factorization is a useful tool in the computation of some optimal control problems, such as algebraic Riccati equation. In this paper, combining the block matrix-vector equation approach with the technique of Lyapunov control function and the Banach fixed point principle, we obtain some new rigorous perturbation bounds and first-order perturbation bound for the factor R of the SR factorization under normwise perturbations, which improved the existing results.New criteria for nonsingular \(H\)-matrices.https://zbmath.org/1449.150792021-01-08T12:24:00+00:00"Liu, Changtai"https://zbmath.org/authors/?q=ai:liu.changtai"Xu, Jing"https://zbmath.org/authors/?q=ai:xu.jing"Xu, Huijun"https://zbmath.org/authors/?q=ai:xu.huijunSummary: To determine a given matrix being a nonsingular \(H\)-matrix or not plays an important role in mathematical economics, control theory, and so on. To get more nonsingular \(H\)-matrices easily, several practical sufficient conditions for nonsingular \(H\)-matrices are obtained by constructing exquisite positive diagonal matrices and applying some techniques of inequalities. The corresponding results are improved and extended. Advantages of these results are illustrated by a numerical example.Computing all Laplacian H-eigenvalues for a uniform loose path of length three.https://zbmath.org/1449.051952021-01-08T12:24:00+00:00"Yue, Junjie"https://zbmath.org/authors/?q=ai:yue.junjie"Zhang, Liping"https://zbmath.org/authors/?q=ai:zhang.lipingSummary: The spectral theory of Laplacian tensor is an important tool for revealing some important properties of a hypergraph. It is meaningful to compute all Laplacian H-eigenvalues for some special \(k\)-uniform hypergraphs. For a \(k\)-uniform loose path of length three, the Laplacian H-spectrum has been studied when \(k\) is odd. However, all Laplacian H-eigenvalues of a \(k\)-uniform loose path of length three have not been found out. In this paper, we compute all Laplacian H-eigenvalues for a \(k\)-uniform loose path of length three. We show that the number of Laplacian H-eigenvalues of an odd(even)-uniform loose path with length three is 7(14). Some numerical results are given to show the efficiency of our method. Especially, the numerical results show that its Laplacian H-spectrum converges to \(\{0, 1, 1.5, 2\}\) when \(k\) goes to infinity. Finally, we show that the convergence of the Laplacian H-spectrum from theoretical analysis.On eigenvalues of generalized shift linear vector isomorphisms.https://zbmath.org/1449.150242021-01-08T12:24:00+00:00"Shirazi, Fatemeh Ayatollah Zadeh"https://zbmath.org/authors/?q=ai:shirazi.fatemeh-ayatollah-zadeh"Soleimani, Elham"https://zbmath.org/authors/?q=ai:soleimani.elhamSummary: Our main aim is to compute eigenvalues of generalized shift isomorphism \(\sigma_\varphi:V^\Gamma\to V^\Gamma\) with \(\sigma_\varphi((x_\alpha)_{\alpha\in\Gamma})=(x_{\varphi(\alpha)})_{\alpha\in\Gamma} ((x_\alpha)_{\alpha\in\Gamma}\in V^\Gamma)\) where \(V\) is a
vector space (over field \(F\)), \(\Gamma\) is a nonempty arbitrary set and \(\varphi:\Gamma\to\Gamma\) is an arbitrary bijection.Real representation and inverse matrix of elliptic commutative quaternion matrix.https://zbmath.org/1449.150742021-01-08T12:24:00+00:00"Kong, Xiangqiang"https://zbmath.org/authors/?q=ai:kong.xiangqiangSummary: Based on the introduction of elliptic commutative quaternion, firstly, it is proved that the elliptic commutative quaternion and the 4-order matrix on the real field are isomorphic. The study of the elliptic commutative quaternion is transformed into the study of the 4-order matrix on the real field. Secondly, using the real representation of the elliptic commutative quaternion matrix, the study of the elliptic commutative quaternion matrix is transformed into the study of the \(4n\)-order matrix on the real field. A series of important properties of real representation of elliptic commutative quaternion matrix are obtained. Finally, based on the real representation properties, the sufficient and necessary conditions for the existence of the eigenvalues of the elliptic commutative quaternion matrix are obtained. The method to find the inverse matrix of the elliptic commutative quaternion matrix is given. And the correctness of the result is verified by a numerical example.On tensor products of Boolean semi-modules and their term rank preservers.https://zbmath.org/1449.150682021-01-08T12:24:00+00:00"Sassanapitax, L."https://zbmath.org/authors/?q=ai:sassanapitax.lee"Pianskool, S."https://zbmath.org/authors/?q=ai:pianskool.sajee"Siraworakun, A."https://zbmath.org/authors/?q=ai:siraworakun.aSummary: We introduce the notion of term rank of an element in tensor products of Boolean semi-modules. We also characterize the linear transformation which general-izes the classical term ranks preserving the Boolean matrices.New construction of deterministic compressed sensing matrices based on vector space over finite fields.https://zbmath.org/1449.150762021-01-08T12:24:00+00:00"Liu, Xuemei"https://zbmath.org/authors/?q=ai:liu.xuemei"Fan, Qianyu"https://zbmath.org/authors/?q=ai:fan.qianyu"Sheng, Shouqiong"https://zbmath.org/authors/?q=ai:sheng.shouqiongSummary: A compressed sensing matrix based on vector spaces over finite fields is constructed, and the coherence of the matrix is computed. A comparison is made with the matrix constructed based on polynomials over finite fields. The character of compressing and recovering signals is better than that of the matrix constructed based on polynomials over finite fields when the matrix satisfies some conditions. The favorable performance of the matrix is demonstrated by numerical simulations.The parallel sum for adjointable operators on Hilbert \({C^*}\)-modules.https://zbmath.org/1449.460492021-01-08T12:24:00+00:00"Luo, Wei"https://zbmath.org/authors/?q=ai:luo.wei"Song, Chuanning"https://zbmath.org/authors/?q=ai:song.chuanning"Xu, Qingxiang"https://zbmath.org/authors/?q=ai:xu.qingxiangSummary: The parallel sum for adjointable operators on Hilbert \({C^*}\)-modules is introduced and studied. Some results known for matrices and bounded linear operators on Hilbert spaces are generalized to the case of adjointable operators on Hilbert \({C^*}\)-modules. It is shown that there exist a Hilbert \({C^*}\)-module \(H\) and two positive operators \(A, B \in \mathcal{L}(H )\) such that the operator equation \({A^{1/2}} = {({A + B} )^{1/2}}X\), \(X \in \mathcal{L}(H )\) has no solution, where \(\mathcal{L}(H )\) denotes the set of all adjointable operators on \(H\).A bound for the rank-one transient of inhomogeneous matrix products in special case.https://zbmath.org/1449.150642021-01-08T12:24:00+00:00"Kennedy-Cochran-Patrick, Arthur"https://zbmath.org/authors/?q=ai:kennedy-cochran-patrick.arthur"Sergeev, Sergeĭ"https://zbmath.org/authors/?q=ai:sergeev.sergei-alekseevich|sergeev.sergei-m|sergeev.sergei-n"Berežný, Štefan"https://zbmath.org/authors/?q=ai:berezny.stefanSummary: We consider inhomogeneous matrix products over max-plus algebra, where the matrices in the product satisfy certain assumptions under which the matrix products of sufficient length are rank-one, as it was shown in [\textit{L. Shue, B. D. O. Anderson}, and \textit{S. Dey}, ``On steady-state properties of certain max-plus products'', in: Proceedings of the 1998 American Control Conference, Philadelphia, Pensylvania, June 24-26,1998. Piscataway, NJ: IEEE. 1909--1913 (1998; \url{doi:10.1109/acc.1998.707354}]. We establish a bound on the transient after which any product of matrices whose length exceeds that bound becomes rank-one.Fast approximate truncated SVD.https://zbmath.org/1449.650822021-01-08T12:24:00+00:00"Shishkin, Serge L."https://zbmath.org/authors/?q=ai:shishkin.serge-l"Shalaginov, Arkadi"https://zbmath.org/authors/?q=ai:shalaginov.arkadi"Bopardikar, Shaunak D."https://zbmath.org/authors/?q=ai:bopardikar.shaunak-dThe paper proposes a new deterministic method for truncated singular value decomposition (SVD) of an arbitrary matrix with predetermined approximation rank \(r\), and with rigorous error control. The method has several important features, as follows. It does not use randomization, or any special structure of the matrix. Asymptotic estimate of the operation count is the same as the one for existing deterministic SVD methods for symmetric matrices, and is lower than the one for existing deterministic methods for unsymmetric matrices. On the other hand, when compared to the randomized algorithms this estimate is slightly higher, but it has better error estimate and is exact in case of full SVD. There are two new features of the method that enable such improvements. The orthogonal matrix of singular vectors is computed in reverse order which is crucial for low computational costs, and intermediate factorizations of higher rank than \(r\) are computed, which increase approximation accuracy. Besides that, the method is one-pass, meaning that each entry of the matrix is reached only once, and is suitable for parallelization.
Majority of the paper is devoted to description of the method as a divide-and-conquer scheme, deriving computational costs and estimating its accuracy. The emphasis was put on theoretical justification of the new method, rather than the development of highly optimized software. Even numerical tests at the end illustrate only its accuracy, and compare it with a randomized algorithm. The tests show that the approximation error vanishes as \(r\) grows, long before it reaches the rank of the input matrix, and confirm its superiority over randomized algorithms.
Reviewer: Nela Bosner (Zagreb)New eigenvalue localizations of quaternionic matrices.https://zbmath.org/1449.150302021-01-08T12:24:00+00:00"Yin, Caixia"https://zbmath.org/authors/?q=ai:yin.caixia"Li, Chaoqian"https://zbmath.org/authors/?q=ai:li.chaoqianSummary: For the eigenvalue localization problem of quaternionic matrices, a new set for locating left and right eigenvalues of quaternionic matrices is obtained, which improves some existing results. An example is given to illustrate the effectiveness of the results.On the square root of quadratic matrices.https://zbmath.org/1449.150452021-01-08T12:24:00+00:00"Zardadi, A."https://zbmath.org/authors/?q=ai:zardadi.akramSummary: Here we present a new approach to calculating the square root of a quadratic matrix. Actually, the purpose of this article is to show how the Cayley-Hamilton theorem may be used to determine an explicit formula for all the square roots of \(2\times 2\) matrices.Iterative algorithm with linear convergence rate for determining strong \(\boldsymbol{H}\)-tensors.https://zbmath.org/1449.650612021-01-08T12:24:00+00:00"Liu, Rui"https://zbmath.org/authors/?q=ai:liu.rui"Liu, Qilong"https://zbmath.org/authors/?q=ai:liu.qilong"Chen, Zhen"https://zbmath.org/authors/?q=ai:chen.zhenSummary: Based on the higher-order power method for computing the spectral radius of nonnegative tensors, we proposed a new iterative algorithm for determining strong \(\boldsymbol{H}\)-tensors. We proved that the given algorithm stops in finite steps and its convergent rate is linear by combined with the scaling technique of inequality and Perron-Frobenius theorem of nonnegative tensors. Some numerical examples show that the algorithm can determine whether a given tensor is a strong \(\boldsymbol{H}\)-tensor or not. The iterative steps of the algorithm are less than that of the classical algorithm for determining strong \(\boldsymbol{H}\)-tensors in some cases.Spectral condition-number estimation of large sparse matrices.https://zbmath.org/1449.650972021-01-08T12:24:00+00:00"Avron, Haim"https://zbmath.org/authors/?q=ai:avron.haim"Druinsky, Alex"https://zbmath.org/authors/?q=ai:druinsky.alex"Toledo, Sivan"https://zbmath.org/authors/?q=ai:toledo.sivanSummary: We describe a randomized Krylov-subspace method for estimating the spectral condition number of a real matrix \(A\) or indicating that it is numerically rank deficient. The main difficulty in estimating the condition number is the estimation of the smallest singular value \(\sigma_{\min}\) of \(A\). Our method estimates this value by solving a consistent linear least squares problem with a known solution using a specific Krylov-subspace method called LSQR. In this method, the forward error tends to concentrate in the direction of a right singular vector corresponding to \(\sigma_{\min}\). Extensive experiments show that the method is able to estimate well the condition number of a wide array of matrices. It can sometimes estimate the condition number when running dense singular value decomposition would be impractical due to the computational cost or the memory requirements. The method uses very little memory (it inherits this property from LSQR), and it works equally well on square and rectangular matrices.Eigenvector sensitivity under general and structured perturbations of tridiagonal Toeplitz-type matrices.https://zbmath.org/1449.150712021-01-08T12:24:00+00:00"Noschese, Silvia"https://zbmath.org/authors/?q=ai:noschese.silvia"Reichel, Lothar"https://zbmath.org/authors/?q=ai:reichel.lotharThe authors start with the remark that the sensitivity of the eigenvalues (in contrast to the eigenvectors) to perturbations of the matrix entries has been well researched. The sensitivity of the eigenvalues is measured by the pseudospectrum.
The main aim of the authors consists in the investigation of the sensitivity of the eigenvectors of (I) tridiagonal Toeplitz matrices to perturbations which are given by changing the first and last diagonal entries of the matrix, and of (II) tridiagonal Toeplitz-type matrices to perturbations that are obtained by modifying the first and last entries of the principal diagonal of the tridiagonal Toeplitz matrix by two parameters. Both kinds of matrices especially arise in numerous computational applications of differential equations.
Formulas for the eigenvalues and eigenvectors of tridiagonal Toeplitz matrices are well-known. The eigenvalues and eigenvectors of the tridiagonal Toeplitz-type matrices are known in closed form for several choices of the two parameters. Based on these formulas, the authors develop minimal eigenvalue distances between two eigenvalues, and eigenvalue condition numbers, in order to discuss the sensitivity of the eigenvalues to general unstructured perturbations of the matrices.
Next, eigenvector condition numbers are derived and the eigenvalue and eigenvector sensitivity to structured perturbations are discussed.
It is found that the sensitivity of the mentioned eigenvectors to perturbations is greater if the matrix is far from normal. But the eigenvectors are rather insensitive to structured perturbations if the Toeplitz matrix has a special structure, for instance, if the matrix is Hermitian.
Finally, the authors apply the theory to approximate (I) the eigenvalues of real Hermitian tridiagonal matrices using the known eigenvalues of the closest symmetric tridiagonal Toeplitz matrix, and (II) the eigenvectors of non-Hermitian nearly tridiagonal Toeplitz matrices using the known spectral factorization of the closest tridiagonal Toeplitz matrix.
The authors mention that such symmetric tridiagonal matrices arise after \(n\) steps of the symmetric Lanczos algorithm applied to large symmetric matrices, and refer for the corresponding algorithm to [\textit{G. H. Golub} and \textit{C. F. Van Loan}, Matrix Computations, 4th ed., John Hopkins Studies in Mathematical Sciences. Baltimore, MD: The John Hopkins University Press (2013; Zbl 1268.65037)].
Reviewer: Georg Hebermehl (Berlin)New results of Drazin inverse for sum of matrices.https://zbmath.org/1449.150072021-01-08T12:24:00+00:00"Yang, Xiaoying"https://zbmath.org/authors/?q=ai:yang.xiaoying"Liu, Xin"https://zbmath.org/authors/?q=ai:liu.xin.2"Wang, Yaqiang"https://zbmath.org/authors/?q=ai:wang.yaqiangSummary: In this paper, new formulas for the Drazin inverse of the sum of two matrices are given under some conditions which are more general than those used in some current literature. Then, according to ideas of the matrix decomposition and matrix splitting, using the theory of Drazin inverse, new representation of partitioned matrices \(M = \begin{pmatrix}A&B\\ C&D\end{pmatrix}\) is given under some conditions.A new Gaussian Fibonacci matrices and its applications.https://zbmath.org/1449.110342021-01-08T12:24:00+00:00"Prasad, B."https://zbmath.org/authors/?q=ai:prasad.baleshwar|prasad.b-s-r-v|prasad.bhikhari|prasad.baji-nath|prasad.b-v-s-s-s|prasad.bandhu|prasad.bhagwati|prasad.b-k-raghu|prasad.biren|prasad.bhagwat|prasad.brij-nandan|prasad.birendra|prasad.b-e|prasad.baij-nath|prasad.b-g|prasad.b-v-n-s|prasad.bhagwan|prasad.b-jaya|prasad.b-d-c-n|prasad.bhanu|prasad.b-v-l-s|prasad.b-a|prasad.b-s-v|prasad.banu|prasad.b-hari|prasad.b-r-guruSummary: In this paper, we introduced a new Gaussian Fibonacci matrix, \(G^n\) whose elements are Gaussian Fibonacci numbers and we developed a new coding and decoding method followed from this Gaussian Fibonacci matrix, \(G^n\). We established the relations between the code matrix elements, error detection and correction for this coding theory. Correction ability of this method is 93.33\%.Mixed-type reverse order laws associated to \(\{1, 3, 4\}\)-inverse.https://zbmath.org/1449.470072021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Deng, Chunyuan"https://zbmath.org/authors/?q=ai:deng.chunyuanSummary: In this paper, we study the mixed-type reverse order laws to \(\{1, 3, 4\}\)-inverses for closed range operators \(A, B\) and \(AB\). It is shown that \(B\{1, 3, 4\}A\{1, 3, 4\} \subseteq (AB)\{1, 3\}\) if and only if \(R({A^*}AB) \subseteq R (B)\). For every \(A^{(134)} \in A (1, 3, 4)\), we have \( (A^{(134)}AB)\{1, 3, 4\}A\{1, 3, 4\} = (AB)\{1, 3, 4\}\) if and only if \(R (A{A^*}AB) \subseteq R (AB)\). As an application of our results, some new characterizations of the mixed-type reverse order laws associated to the Moore-Penrose inverse and the \(\{1, 3, 4\}\)-inverse are established.Spectral distribution in the eigenvalues sequence of products of \(g\)-Toeplitz structures.https://zbmath.org/1449.150162021-01-08T12:24:00+00:00"Ngondiep, Eric"https://zbmath.org/authors/?q=ai:ngondiep.ericSummary: Starting from the definition of an \(n\times n\) \(g\)-Toeplitz matrix,
\[
T_{n, g}\left (u \right) = [{\hat u}_{r - gs}]_{r, s = 0}^{n - 1},
\]
where \(g\) is a given nonnegative parameter, \(\left\{ {{{\hat u}_k}} \right\}\) is the sequence of Fourier coefficients of the Lebesgue integrable function \(u\) defined over the domain \(\mathbb{T} = \left ({ - \pi, \pi} \right]\), we consider the product of \(g\)-Toeplitz sequences of matrices \(\left\{ {{T_n}\left ({{f_1}} \right){T_n}\left ({{f_2}} \right)} \right\}\), which extends the product of Toeplitz structures \(\left\{ {{T_n}\left ({{f_1}} \right){T_n}\left ({{f_2}} \right)} \right\}\), in the case where the symbols \({f_1}, {f_2} \in {L^\infty}\left (\mathbb{T} \right)\). Under suitable assumptions, the spectral distribution in the eigenvalues sequence is completely characterized for the products of \(g\)-Toeplitz structures. Specifically, for \(g \ge 2\) our result shows that the sequences \(\left\{ {{T_{n, g}}\left ({{f_1}} \right){T_{n, g}}\left ({{f_2}} \right)} \right\}\) are clustered to zero. This extends the well-known result, which concerns the classical case (that is, \(g = 1\)) of products of Toeplitz matrices. Finally, a large set of numerical examples confirming the theoretic analysis is presented and discussed.On the permanental sum of bicyclic graphs.https://zbmath.org/1449.051442021-01-08T12:24:00+00:00"Wu, Tingzeng"https://zbmath.org/authors/?q=ai:wu.tingzeng"Das, Kinkar Chandra"https://zbmath.org/authors/?q=ai:das.kinkar-chandraSummary: Let \(A(G)\) be the adjacency matrix of a graph \(G\). The permanental polynomial of \(G\) is defined as \(\pi (G,x)=\mathrm{per}(xI-A(G))\). The permanental sum of \(G\) can be defined as the sum of the absolute values of the coefficients of \(\pi (G,x)\). In this paper, we investigate the properties of the permanental sum of bicyclic graphs. We present upper and lower bounds of the permanental sum of bicyclic graphs, and the corresponding extremal bicyclic graphs are also determined.\(S\)-type bounds for the largest singular value of nonnegative rectangular tensors.https://zbmath.org/1449.150232021-01-08T12:24:00+00:00"Sang, Caili"https://zbmath.org/authors/?q=ai:sang.caili"Zhao, Jianxing"https://zbmath.org/authors/?q=ai:zhao.jianxingSummary: For estimates of the largest singular value of a non-negative rectangular tensor \(A\), by breaking \(N\) into disjoint nonempty proper subsets \(S\) and its complement \(\bar{S}\), and using classification discussion idea and some techniques of inequalities, \(S\)-type upper and lower bounds for the largest singular value of \(A\) are presented and proved to be an improvement of some existing results. Numerical examples are given to verify the theoretical results which show that the new bounds are more accurate than existing sets.Third Hankel determinant for Ma-Minda bi-univalent functions.https://zbmath.org/1449.300342021-01-08T12:24:00+00:00"Zhang, Haiyan"https://zbmath.org/authors/?q=ai:zhang.haiyan"Tang, Huo"https://zbmath.org/authors/?q=ai:tang.huoSummary: In this paper, we investigate the third Hankel determinant \({H_3} (1)\) for the class \(H_\sigma^\mu (\lambda, \varphi)(\lambda \ge 1, \mu \ge 1)\) of Ma-Minda bi-univalent functions in the open unit disk \(\mathbb{D} = \{ z:| z | < 1\}\) and obtain the upper bound of the above determinant \({H_3} (1)\).Brauer upper bound for the \(Z\)-spectral radius of nonnegative tensors.https://zbmath.org/1449.150492021-01-08T12:24:00+00:00"He, Jun"https://zbmath.org/authors/?q=ai:he.jun"Ke, Hua"https://zbmath.org/authors/?q=ai:ke.hua"Liu, Yanmin"https://zbmath.org/authors/?q=ai:liu.yanmin"Tian, Junkang"https://zbmath.org/authors/?q=ai:tian.junkangSummary: In this paper, we propose an upper bound for the largest \(Z\)-eigenvalue of an irreducible weakly symmetric and nonnegative tensor, which is called the Brauer upper bound: \[\rho_Z (A)\le\frac{1}{2}\max\limits_{\begin{subarray} {1}{i,j \in N}\\ {j \ne i}\end{subarray}} ({a_{i\cdots i}} + {a_{j\cdots j}} + \sqrt{({a_{i\cdots i}}-{a_{j\cdots j}})^2 + 4{r_i} (A){r_j} (A)}),\] where \({r_i} (A) = \sum\limits_{{ii_2} \cdots {i_m} \ne ii \cdots i} {a_{{ii_2} \cdots {i_m}}}\), \(i, {i_2}, \dots, {i_m} \in N = \{1,2,\ldots, n\} \). As applications, a bound on the \(Z\)-spectral radius of uniform hypergraphs is presented.Sequences of lower bounds for the spectral radius of Hadamard product of nonnegative matrices.https://zbmath.org/1449.150542021-01-08T12:24:00+00:00"Liu, Hui"https://zbmath.org/authors/?q=ai:liu.hui.3|liu.hui.1|liu.hui.2|liu.hui.4"Huang, Kuanna"https://zbmath.org/authors/?q=ai:huang.kuannaSummary: For the lower bounds of the spectral radius \(\rho (A \circ B)\) of nonnegative matrices \(A\) and \(B\), three monotone increasing and convergent sequences of lower bounds are obtained. The method can easily and tightly get the better bounds. Finally, a numerical example is given to verify the theoretical results and the true value of the spectral radius could be reached in some cases.Independence and matching numbers of unicyclic graphs from null space.https://zbmath.org/1449.051652021-01-08T12:24:00+00:00"Allem, L. Emilio"https://zbmath.org/authors/?q=ai:allem.luiz-emilio"Jaume, Daniel A."https://zbmath.org/authors/?q=ai:jaume.daniel-a"Molina, Gonzalo"https://zbmath.org/authors/?q=ai:molina.gonzalo"Toledo, Maikon M."https://zbmath.org/authors/?q=ai:toledo.maikon-m"Trevisan, Vilmar"https://zbmath.org/authors/?q=ai:trevisan.vilmarSummary: We characterize unicyclic graphs that are singular using the support of the null space of their pendant trees. From this, we obtain closed formulas for the independence and matching numbers of a unicyclic graph, based on the support of its subtrees. These formulas allows one to compute independence and matching numbers of unicyclic graphs using linear algebra methods.Eigenvalue localization and determination of non-singularity for matrices.https://zbmath.org/1449.150192021-01-08T12:24:00+00:00"Sang, Caili"https://zbmath.org/authors/?q=ai:sang.caili"Zhao, Jianxing"https://zbmath.org/authors/?q=ai:zhao.jianxingSummary: By splitting a complex square matrix \(A\) into \(A = sI - B\), where \(s\) is an arbitrary complex number, \(I\) is the identity matrix and \(B\) is a complex square matrix, and by using an existing method of determination of non-singularity for matrices, some eigenvalue inclusion sets and some methods of determination of non-singularity for \(A\) with two parameters (\(s\) and a positive integer \(k\)) are obtained and proved to be more accurate and more general than some existing results. Numerical results show that by adjusting \(s\) and \(k\), the eigenvalues of \(A\) can be located more accurate, and the non-singularity of \(A\) can be determined.On the singular values of the Hankel matrix with application in singular spectrum analysis.https://zbmath.org/1449.150152021-01-08T12:24:00+00:00"Mahmoudvand, Rahim"https://zbmath.org/authors/?q=ai:mahmoudvand.rahim"Zokaei, Mohammad"https://zbmath.org/authors/?q=ai:zokaei.mohammadSummary: Hankel matrices are an important family of matrices that play a fundamental role in diverse fields of study, such as computer science, engineering, mathematics and statistics. In this paper, we study the behavior of the singular values of the Hankel matrix by changing its dimension. In addition, as an application, we use the obtained results for choosing the optimal values of the parameters of singular spectrum analysis, which is a powerful technique in time series based on the Hankel matrix.New upper bounds on the largest eigenvalues of the Hadamard product of nonnegative matrices.https://zbmath.org/1449.150482021-01-08T12:24:00+00:00"Chen, Fubin"https://zbmath.org/authors/?q=ai:chen.fubinSummary: In this paper, for the upper bounds on the largest eigenvalue of the Hadamard product of two nonnegative matrices \(A\) and \(B\), some new estimating formulas are given by using Gerschgorin theorem and Brauer theorem. We compare the new results with the other existing results. Numerical example shows that the new results improve some existing ones in some cases.Orthogonal incremental non-negative matrix factorization algorithm and its application in image classification.https://zbmath.org/1449.680622021-01-08T12:24:00+00:00"Ge, Shaodi"https://zbmath.org/authors/?q=ai:ge.shaodi"Luo, Liuhong"https://zbmath.org/authors/?q=ai:luo.liuhong"Li, Hongjun"https://zbmath.org/authors/?q=ai:li.hongjunSummary: To improve the sparseness of the base matrix in incremental non-negative matrix factorization, we in this paper present a new method, orthogonal incremental non-negative matrix factorization algorithm (OINMF), which combines the orthogonality constraint with incremental learning. OINMF adopts batch update in the process of incremental learning, and its iterative formulae are obtained using the gradient on the Stiefel manifold. The experiments on image classification show that the proposed method achieves much better sparseness and orthogonality, while retaining time efficiency of incremental learning.New sufficient conditions for nonsingular \(H\)-matrices.https://zbmath.org/1449.150102021-01-08T12:24:00+00:00"Tuo, Qing"https://zbmath.org/authors/?q=ai:tuo.qingSummary: In this paper, several new sufficient conditions for nonsingular \(H\)-matrices were given by constructing new positive diagonal factors, which improve some related results. And then the advantages of these sufficient conditions were illustrated by some numerical examples.Some further results on the Smith form of bivariate polynomial matrices.https://zbmath.org/1449.150332021-01-08T12:24:00+00:00"Li, Dongmei"https://zbmath.org/authors/?q=ai:li.dongmei"Liang, Rui"https://zbmath.org/authors/?q=ai:liang.rui"Liu, Jinwang"https://zbmath.org/authors/?q=ai:liu.jinwangSummary: The equivalence of systems plays an important role in 2D systems, which is closely related to the equivalence of bivariate polynomial matrices. In this paper, the Smith forms of several classes of bivariate polynomial matrices are investigated. Some new results and criteria on the equivalence of these matrices are obtained. These criteria are easily checked by computing the reduced Gröbner basis of the ideal generated by the minors of lower order of the given matrix.