Symmetric, positive semidefinite, and positive definite real solutions of \(AX=XA^ T\) and \(AX=YB\) (Q1183139)

An inverse problem of linear optimal control requires the solutions of the equations \(AX=XA^ T\) and \(AX=YB\) with arbitrary nonzero real matrices \(A\) and \(B\) of the same size. The authors seek all real solutions \(X\) and \(Y\) which are: (1) symmetric, (2) symmetric and positive semidefinite, and (3) symmetric and positive definite. They obtain necessary and sufficient conditions for the existence of such solutions and then the general forms of solutions. The basic mathematical tool used is the Jordan normal form for matrices. The results and calculations are too complicated to be reproduced here.