Diagonal and monomial solutions of the matrix equation \(AXB=C\) (Q6486725)

Consider the matrix equation \(AXB=C\), where \(A\), \(B\) and \(C\) are real matrices with appropriate dimensions and \(X\) is an unknown square matrix. The author gives a necessary and sufficient condition under which this equation has a diagonal solution and a monomial solution, respectively. A square matrix is monomial if each row and column contains at most one nonzero entry. The author also presents explicit formulas for these solutions and solves the least squares problem \(\min_X\|C-AXB\|_F\) over diagonal and, respectively, monomial matrices. Here \(\|\cdot\|_F\) denotes the Frobenius norm.