## 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.
Reviewer: V.P.Kostov (Nice)

### MSC:

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