zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Rank equalities and inequalities for Kronecker products of matrices with applications. (English) Zbl 1041.15016
Let $A_1,\dots ,A_k$ be square matrices of orders $n_1,\dots ,n_k$, respectively, and denote $A=A_1\otimes\cdots\otimes A_k$, $n=n_1\dots n_k$, where $A\otimes B$ is the Kronecker product of $A=(a_{ij})_{m\times n}$ and $B=(b_{ij})_{p\times q}$ over a field ${\Bbb F}$, i.e., $A\otimes B=(a_{ij}B)\in{\Bbb F}^{mp\times nq}$. Several rank equalities and inequalities are established. The obtained results are applied to find upper and lower bounds for the dimension of the range of the linear transformations $T_1(X)=X-AXB$ and $T_2(X)=AX-XB.$ The following inequality is proved. Theorem: Let $A=A_1\otimes\cdots\otimes A_k,$ and denote $rk(I_{n_i}-A_i)=r_i$, $i=1,\dots ,k$. Then $rk(I_n-A)\leq n-(n_1-r_1)(n_2-r_2)\dots (n_k-r_k)$.

15A45Miscellaneous inequalities involving matrices
15A03Vector spaces, linear dependence, rank
15A04Linear transformations, semilinear transformations (linear algebra)
Full Text: DOI
[1] Fialkow, L. A.: A note on the operator x\toax-XB. Trans. amer. Math. soc. 243, 147-168 (1978) · Zbl 0397.47002
[2] Fialkow, L. A.: A note on the range of the operator x\toax-XB. Illinois J. Math. 25, 112-124 (1981) · Zbl 0443.47003
[3] Fialkow, L. A.: Essential spectra of elementary operators. Trans. amer. Math. soc. 267, 157-174 (1981) · Zbl 0475.47002
[4] Heinig, G.: The group inverse of the transformation $S(X)=$AX-XB. Linear algebra appl. 257, 321-342 (1997) · Zbl 0874.15004
[5] Horn, R. A.; Johnson, C. R.: Topics in matrix analysis. (1991) · Zbl 0729.15001
[6] Marsaglia, G.; Styan, G. P. H.: Equalities and inequalities for ranks of matrices. Linear multilinear algebra 2, 269-292 (1974) · Zbl 0297.15003
[7] Mazouz, A.: On the range and the kernel of the operator x\mapstoaxb-X. Proc. amer. Math. soc. 127, 2105-2107 (1999) · Zbl 0922.47030
[8] Neudecker, H.: A note on Kronecker products and matrix equation systems. SIAM J. Appl. math. 17, 603-606 (1969) · Zbl 0185.08204
[9] Tian, Y.: How to characterize equalities for the Moore--Penrose inverse of a matrix. Kyungpook math. J. 41, 1-15 (2001) · Zbl 0987.15001
[10] Tian, Y.; Styan, G. P. H.: Rank equalities for idempotent and involutory matrices. Linear algebra appl. 335, 101-117 (2001) · Zbl 0988.15002
[11] Trampus, A.: A canonical basis for the matrix transformation x\toax+XB. J. math. Anal. appl. 14, 242-252 (1966) · Zbl 0145.25205
[12] Van Loan, C. F.: The ubiquitous Kronecker product. J. comput. Appl. math. 123, 85-100 (2000) · Zbl 0966.65039
[13] Zhang, G.: On the operators x\toax-XB and x\toaxb-X. J. fudan univ. Nat. sci. 28, 148-154 (1989)