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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 Ω a finite dimensional central algebra over a field F with an involution σ (charΩ2), and by Ω m×n the set of all m×n-matrices over Ω. For A=(a ij )Ω m×n set A * =(σ(a ji ))Ω n×m , A (*) =(σ(a m-j+1,n-i+1 ))Ω n×m , A =(a m-i+1,n-j+1 )Ω m×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 =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 Ω[λ]:(*)A i X-YB i =C i , (**) A i XB i -C i XD i =E i , i=1,...,s. Necessary and sufficient conditions are given for the existence of bi(skew)symmetric solutions to (*) and (**) over Ω 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:
15A24Matrix equations and identities
15A33Matrices over special rings