DMS Linear Algebra/Algebra Seminar

Speaker: Zhuoheng He

Title: The general \(\phi\)-Hermitian solution to mixed pairs of quaternion matrix Sylvester equations

Abstract: In this talk, we consider two systems of mixed pairs of quaternion matrix Sylvester equations

\[A_{1}X-YB_{1}=C_{1},~A_{2}Z-YB_{2}=C_{2}\] and \[A_{1}X-YB_{1}=C_{1},~A_{2}Y-ZB_{2}=C_{2},\]

where \(Z\) is \(\phi\)-Hermitian. Some practical necessary and sufficient conditions for the existence of a solution \((X,Y,Z)\) to those systems in terms of the ranks and Moore-Penrose inverses of the given coefficient matrices will be presented. Moreover, the general solutions to these systems are explicitly given when they are solvable. We also provide some numerical examples to illustrate our results.