On the singular vectors of the generalized Lyapunov operator

This paper, we study the largest and the smallest singular vectors of the generalized Lyapunov operator. For real matrices A,B with order n, we prove that max_||X||F=1 ||AXB^T + BXA^T||_F is achieved by a symmetric matrix for n ≤ 3 and give a counterexample for order n = 4. We also prove that min_||X||F=1 ||AXB^T +BXA^T||_F is achieved by a symmetric matrix for n ≤ 2 and give a counterexample for order n = 3. It is shown that the minimizer is symmetric, if the minimum is zero, or if the real parts of the eigenvalues of A−λB are of one sign.