Consistency for bi(skew)symmetric solutions to systems of generalized Sylvester equations over a finite central algebra Original Research Article

Let Ω be a finite dimensional central algebra with an involutorial antiautomorphism σ and char Ω≠2, Ωn×n be the set of all n×n matrices over Ω. A=(aij)set membership, variantΩn×n is called bisymmetric if aij=an−i+1,n−j+1=σ(aji) and biskewsymmetric if aij=−an−i+1,n−j+1=−σ(aji). The following systems of generalized Sylvester equations over Ω[λ]: image image are considered. Necessary and sufficient conditions are given for the existence of constant solutions with bi(skew)symmetric constrains to (I) and (II). As a particular case, auxiliary results dealing with the system of Sylvester equations are also presented.