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

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 positive-definite hermitian.

Recently, algebraists too have become interested in stable matrices.Definition: A square matrix is stable all its characteristic values have negative real parts.
(In this article, the entries of all matrices are complex numbers unless stated otherwise.) A classical test for stability of matrices is Lyapunov's theorem, whose statement is facilitated by some notation: S set of all nxn stable matrices H set of all nxn hermitian matrices i H set of all nxn skew-hermitian matrices H set of all nxn positive-definite hermitian matrices N set of all nxn negative-definite hermitian matrices fA(X) AX+XA*, where A and X are nxn matrices and A* is the con- jugate transpose of A.
(It is trivial to verify that fA('), the Lyapunov mapping, is a linear transformation on the linear space M n of nxn matrices.) Lyapunov's theorem is usually expressed as statement a) of Theorem I: The following three statements are equivalent: a) A S there exists G H such that A(G) -I; T. HAYNES b) A S for every Glen there exists G H such that fA(G) G I there exists GlNand there exists G N such that fA(G) G I [Taussky, 1964;p. 6, thms 2-3]; c) Let C al+S (a real and < 0, S i H) and D diag(d I dn) with d i real (i=l n).Then CD $ d i > 0 for all i. [Taussky, 1961, J. Math Anal.& App.].
The equivalences are proven (essentially) in Taussky's articles.An analytic proof a) is in Bellman, pp.242-245, and a topological proof in Ostrowski & Schneider.
Some useful theorems result if further restrictions are imposed on A besides stability.These theorems are obtained via a topological route and require additional concepts.
Definition: The inertia of an nxn matrix X is the ordered triple of integers ((X), (X), (X)) In(X) where (X) is the number of characteris- tic values of X whose real parts are positive, (X) the number whose real parts are negative, and (X) the number whose real parts are 0. If nxn matrices M and N possess the same inertia, this will be denoted by MN.
Let M and N be nxn hermitian matrices.M and N are congruent (denoted M N) 3 P non-singular such that M P*NP.
Recall that all norms in the set of all nxn matrices M n induce the same topology.In M n so topologized, matrices M and N are connected there exists a connected set containing both M and N. The relationship of being connected is an equivalence relation, which will be denoted by u M and N are arc-wise connected there exists a continuous function f from the real interval [0,I] into M n such that f(0) M and f(1) N. This, too, is an equivalence relation in M n and will be denoted by, a The preceding concepts are brought together by the following theorem: Theorem 3: In the set N n of all non-singular nxn matrices with the relative topology induced by any norm, A UB and Aa B (A, B Nn).
[Schneider" pp. 818-819,lemmata I & 2].Let denote the set of all nxn hermitian matrices of rank r.In with the relative topology induced by any norm the four equivalence relations N,u ,a c coincide.[Schneider" p. 820].
The relationship between algebraic features of hermitian matrices and topological features expressed by theorem 3 makes it possible to discover the variation in signature induced by the Lyapunov mapping fA(') whenever A S is normal and H Since A is normal, it is unitarily similar to a diagonal matrix: VAV* diag(a I an) V unitary.Also a basis for n-dimensional space can be