On the Singular “Vectors” of the Lyapunov Operator

For a real matrix A, the separation of $A^T $ and A is sep $(A^T , - A) = \min \| A^T X + XA \| / \| X \|$, where $\| \cdot \|$ represents the Frobenius matrix norm. We discuss the conjecture that the minimizer X is symmetric. This conjecture is related to the numerical stability of methods for solving the matrix Lyapunov equation. The quotient is minimized by either a symmetric matrix or a skew-symmetric matrix and is maximized by a symmetric matrix. The conjecture is true if A is 2-by-2, if A is normal, if the minimum is zero, or if the real parts of the eigenvalues of A are of one sign. In general the conjecture is false, but counterexamples suggest that symmetric matrices are nearly optimal.