%A Ralph Byers
%A Stephen Nash
%T On the Singular "Vectors" of the Lyapunov Operator
%R Technical Report 438
%I Mathematical Sciences Department,
The Johns Hopkins University
%D June 1985
%X For a real matrix A, the separation of A' and A is sep(A',-A) =
min ||A'X + XA||/||X||, where ||.|| represents the Frobenius norm,
and A' is the transpose of A.  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 counter examples suggest that symmetric matrices are
nearly optimal.