The reflexive and anti-reflexive solutions of the matrix equation \(AX=B\). (Q1414702)

Let \(P\in\mathbb{C}^{n\times n}\) be both Hermitian and unitary. A matrix \(X\in\mathbb{C}^{n\times n}\) is said to be reflexive (resp. anti-reflexive) with respect to \(P\) when \(X= PXP\) (resp. \(X = -PXP\)). Conditions are given for existence of reflexive and antireflexive solutions (with respect to \(P\)) of the equation (1) \(AX= B\), where \(A,B\in\mathbb{C}^{m\times n}\) are given matrices. If \(X_0\in \mathbb{C}^{n\times n}\) is a given matrix, expressions are given for the reflexive and anti-reflexive solutions of (1) which are the nearest to \(X_0\) in the Frobenius norm.