Realizable solutions to matrix polynomial equations \(X(z_ 1)A(z_ 2)+B(z_ 1)Y(z_ 2)=C(z_ 1z_ 2)\) (Q913891)

This paper considers the matrix polynomial equation \(X(z_ 1)A(z_ 2)+B(z_ 1)Y(z_ 2)=C(z_ 1,z_ 2)\) arising in decomposition of 2-D systems into 1-D systems, where \(A(z_ 2)\), \(B(z_ 1)\), \(C(z_ 1,z_ 2)\) are given and \(X(z_ 1)\), \(Y(z_ 2)\) are unknown. By rewriting \(A(z_ 2)\), \(B(z_ 1)\), \(C(z_ 1,z_ 2)\) in their power extension forms as \(A(z_ 2)=\sum^{nA}_{i=0}A_ iz^ i_ 2,\) \(B(z_ 1)=\sum^{mB}_{i=0}B_ iz^ i_ 1,\) \(C(z_ 1,z_ 2)=\sum^{mC}_{i=0}\sum^{nC}_{i=0}C_{ii}z^ i_ 1z^ i_ 2\) the necessary and sufficient conditions for the existence of a realizable solution and a procedure for finding it are established in terms of coefficient matrices \(A_ i\), \(B_ i\), and \(C_{ii}\).