zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra. (English) Zbl 1004.15017
Denote by $\Omega$ a finite dimensional central algebra over a field $F$ with an involution $\sigma$ (char$\Omega \neq 2$), and by $\Omega ^{m\times n}$ the set of all $m\times n$-matrices over $\Omega$. For $A=(a_{ij})\in \Omega ^{m\times n}$ set $A^*=(\sigma (a_{ji}))\in \Omega ^{n\times m}$, $A^{(*)}=(\sigma (a_{m-j+1,n-i+1}))\in \Omega ^{n\times m}$, $A^{\sharp }=(a_{m-i+1,n-j+1})\in \Omega ^{m\times n}$. $A$ is called (skew)selfconjugate if $A=A^*$ (if $A=-A^*$), per(skew)selfconjugate if $A=A^{(*)}$ (if $A=-A^{(*)}$), centro(skew)symmetric if $A^{\sharp }=A$ (if $A^{\sharp }=-A$). Any two of these three properties imply the third one. $A$ is bi(skew)symmetric if it is (skew)selfconjugate and per(skew)selfconjugate at the same time. The paper treats the following system of matrix equations over $\Omega [\lambda ]: (*)\ A_iX-YB_i=C_i$, $(**)$ $A_iXB_i-C_iXD_i=E_i$, $i=1,\ldots ,s$. Necessary and sufficient conditions are given for the existence of bi(skew)symmetric solutions to $(*)$ and $(**)$ over $\Omega$ and of solutions $(X,Y)$ to $(*)$ where $X$ is bisymmetric (biskewsymmetric) and $Y$ is biskewsymmetric (bisymmetric). Auxiliary results dealing with systems of Sylvester equations $AX-XB=C$ or $AX-YB=C$ are also presented.

MSC:
15A24Matrix equations and identities
15B33Matrices over special rings (quaternions, finite fields, etc.)
WorldCat.org
Full Text: DOI
References:
[1] Guralnick, R. M.: Roth’s theorems for sets of matrices. Linear algebra appl. 71, 113-117 (1985) · Zbl 0584.15006
[2] Guralnick, R. M.: Matrix equivalence and isomorphism of modules. Linear algebra appl. 43, 125-136 (1982) · Zbl 0493.16015
[3] Guralnick, R. M.: Roth’s theorems and decomposition of modules. Linear algebra appl. 39, 155-165 (1981) · Zbl 0468.16022
[4] Gustafson, W.; Zelmanowitz, J.: On matrix equivalence and matrix equations. Linear algebra appl. 27, 219-224 (1979) · Zbl 0419.15009
[5] Hartwig, R.: Roth’s equivalence problem in unite regular rings. Proc. amer. Math. soc. 59, 39-44 (1976) · Zbl 0347.15005
[6] Gustafson, W.: Roth’s theorems over commutative rings. Linear algebra appl. 23, 245-251 (1979) · Zbl 0398.15013
[7] Wimmer, H. K.: Roth’s theorems for matrix equations with symmetry constraints. Linear algebra appl. 199, 357-362 (1994) · Zbl 0796.15014
[8] Hartwig, R.: Roth’s removal rule revisited. Linear algebra appl. 49, 91-115 (1984) · Zbl 0509.15004
[9] Huang, L.; Liu, J.: The extension of Roth’s theorem for matrix equations over a ring. Linear algebra appl. 259, 229-235 (1997) · Zbl 0880.15016
[10] Wimmer, H. K.: Consistency of a pair of generalized Sylvester equations. IEEE trans. Automat. control 39, 1014-1015 (1994) · Zbl 0807.93011
[11] Wang, Q. W.; Li, S. Z.: On the center (skew-) self-conjugate solutions to the systems of matrix equations over a finite dimensional central algebra. Math. sci. Res. hot-line 12, 11-17 (2001) · Zbl 1085.15501
[12] Roth, W. E.: The equation AX-YB=C and AX-XB=C in matrices. Proc. amer. Math. soc. 3, 392-396 (1952) · Zbl 0047.01901
[13] K.E. Chu, Exclusion theorems for the generalized eigenvalue problem, Numerical Analysis Report NA/11/85, Department of Mathematics, University of Reading, 1985
[14] Epton, M. A.: Methods for the solution of AXB-CXD=E and its application in the numerical solution of implicit ordinary differential equations. Bit 20, 341-345 (1980) · Zbl 0452.65015
[15] Hernandez, V.; Gasso, M.: Explicit solution of the matrix equation AXB-CXD=E. Linear algebra appl. 121, 333-344 (1989)
[16] Wimmer, H. K.: Linear matrix equations: the module theoretic approach. Linear algebra appl. 120, 149-164 (1989) · Zbl 0677.15001
[17] Huang, L.: The matrix equation AXB-GXD=E over the quaternion field. Linear algebra appl. 234, 197-208 (1996) · Zbl 0840.15017
[18] Cantoni, A.; Butler, P.: Eigenvalues and eigenvectors of symmetric centrosymmetric matrices. Linear algebra appl. 13, 275-288 (1976) · Zbl 0326.15007
[19] Reid, R. M.: Some eigenvalues properties of persymmetric matrices. SIAM rev. 39, 313-316 (1997) · Zbl 0876.15006
[20] Andrew, A. L.: Centrosymmetric matrices. SIAM rev 40, 697-698 (1998) · Zbl 0918.15006
[21] Weaver, J. R.: Centrosymmetric (cross-symmetric) matrices, their basic properties, eigenvalues, eigenvectors. Amer. math monthly 92, 711-717 (1985) · Zbl 0619.15021
[22] Wimmer, H. K.: The matrix equation X-AXB=C and an analogue of Roth’s theorem. Linear algebra appl. 109, 145-147 (1988) · Zbl 0656.15005
[23] Kucera, V.; Zagalak, P.: Constant solutions of polynomial equations. Int. J. Contr. 53, 495-502 (1991) · Zbl 0731.15009
[24] Demmel, J.: Computing stable eigendecompositions of matrix quadruples. Linear algebra appl. 88/89, 139-186 (1987) · Zbl 0627.65032
[25] Drexl, P. K.: Skew field. London mathematical society lecture notes 81 (1983)