The constrained solutions of two matrix equations (Q1565983)

The authors study the symmetric positive semidefinite solution of the matrix equation \(AX_1A^T+BX_2B^T=C\) where \(A\) (\(B\), \(C\)) is \(m\times n\) (\(m\times p\), \(m\times m\)); and the bisymmetric positive semidefinite solution of the matrix equation \(D^TXD=C\) where \(D\) (\(C\)) is \(n\times m\) (\(m\times m\)); all matrices are real. They use generalized singular value decomposition and obtain necessary and sufficient conditions for the existence of solutions as well as formulae for these solutions. (A symmetric matrix \(A\) is called bisymmetric if it is symmetric w.r.t. each of the two diagonals. It is called positive semidefinite if \(Y^TAY\geq 0\) for every real \(n\)-vector \(Y\)).