Changes in signature induced by the Lyapunov mapping \({\mathfrak L}_ A:X\to AX+XA^*\) (Q1123951)

The Lyapunov mapping on \(n\times n\) matrices over C is defined by \({\mathfrak L}_ A(X)=AX+XA^*;\) a matrix is stable iff all its characteristic values have negative real parts; and the inertia of a matrix X is the ordered triple \(In(X)=(\pi,\nu,\delta)\) where \(\pi\) is the number of eigenvalues of X whose real parts are positive, \(\nu\) the number whose real parts are negative, and \(\delta\) the number whose real parts are 0. It is proven that for any normal, stable matrix A and any Hermitian matrix H, if \(In(H)=(\pi,\nu,\delta)\) then \(In({\mathfrak L}_ A(H))=(\nu,\pi,\delta).\) Further, if the stable matrix A has only simple elementary divisors, then the image under \({\mathfrak L}_ A\) of a positive- definite Hermitian matrix is negative-definite Hermitian, and the image of a negative-definite Hermitian matrix is positive-definite Hermitian. These results are mild generalizations of the classical theorem of Lyapunov characterizing stable matrices.