A matrix equation from an inverse problem of vibration theory (Q1812222)

The matrix equation \(XA=YAD\) arises from an inverse problem in vibration theory. Here \(A\neq 0\) is an \(n\times m\) real matrix of rank \(r\), and \(D\neq 0\) is a real symmetric \(m\times m\) one, \(X\) is the stiffness and \(Y\) is the mass matrix of the vibrating system. The authors consider the symmetric, positive semidefinite and positive definite real solutions to the equation. When \(D=D^T\) they derive necessary and sufficient conditions for the existence of such solutions and their general forms.