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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.

15A24Matrix equations and identities
15B33Matrices over special rings (quaternions, finite fields, etc.)
Full Text: DOI
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