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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.
Reviewer: V.P.Kostov (Nice)

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