Backward perturbation analysis for the matrix equation \(A^T X A + B^T Y B = D\) (Q2431056)

The authors consider the linear matrix equation \(A^T X A + B^T Y B = D\), where \(A,B\) are \(n \times n\) real matrices and \(D\) symmetric positive semi-definite matrix. The normwise backward perturbation bounds for the solution of the equation are derived by applying the Brouwer fixed-point theorem and the singular value decomposition as well as the property of the Kronecker product. The results are illustrated by two simple numerical examples.