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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:
 15A24 Matrix equations and identities 15B33 Matrices over special rings (quaternions, finite fields, etc.)
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##### 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)