CHANGES IN SIGNATURE INDUCED BY THE LYAPUNOV MAPPING 2A X AX + XA*

ABSTRACT. The apunov mapping on n x n matrices over C is defined by ZA(X AX + XA* " a matrix is stable iffall its characteristic values have negative real parts " and the inertia of a matrix X is the ordered triple In(X) (,w,6) where is the number of eigenvalues of X whose real parts are positive, the number whose real parts are negative, and 6 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) (,w,6) then In(A(H)) (w,,6). Further, if stable matrix A has only simple elementary divisors, then the image under ZA of a positive-definite hermitian matrix is negative-definite hermitian, and the image of a negative-definite hermitian matrix is posi-tive-definite hermitian. KEY WORDS AND PHRASES. Lyapunov, stable matrix, matrix inertia, positive-definite matrix 1980 AMS SUBJECT CLASSIFICATION CODES. 15A18, 15A42 For many years stable matrices have interested applied mathematician