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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${\Omega }\ne 2$), and by ${{\Omega }}^{m×n}$ the set of all $m×n$-matrices over ${\Omega }$. For $A=\left({a}_{ij}\right)\in {{\Omega }}^{m×n}$ set ${A}^{*}=\left(\sigma \left({a}_{ji}\right)\right)\in {{\Omega }}^{n×m}$, ${A}^{\left(*\right)}=\left(\sigma \left({a}_{m-j+1,n-i+1}\right)\right)\in {{\Omega }}^{n×m}$, ${A}^{♯}=\left({a}_{m-i+1,n-j+1}\right)\in {{\Omega }}^{m×n}$. $A$ is called (skew)selfconjugate if $A={A}^{*}$ (if $A=-{A}^{*}$), per(skew)selfconjugate if $A={A}^{\left(*\right)}$ (if $A=-{A}^{\left(*\right)}$), centro(skew)symmetric if ${A}^{♯}=A$ (if ${A}^{♯}=-A$). 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 ${\Omega }\left[\lambda \right]:\left(*\right)\phantom{\rule{4pt}{0ex}}{A}_{i}X-Y{B}_{i}={C}_{i}$, $\left(**\right)$ ${A}_{i}X{B}_{i}-{C}_{i}X{D}_{i}={E}_{i}$, $i=1,...,s$. Necessary and sufficient conditions are given for the existence of bi(skew)symmetric solutions to $\left(*\right)$ and $\left(**\right)$ over ${\Omega }$ and of solutions $\left(X,Y\right)$ to $\left(*\right)$ 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 15A33 Matrices over special rings