Inertia Theorems for Matrices: the Semi-definite Case

1. The inertia of a square matrix A with complex elements is defined to be the integer triple In A = (ir(A), V(A), 8(4)), where ir(A) {v(A)} equals the number of eigenvalues in the open right {left} half plane, and 8(A) equals the number of eigenvalues on the imaginary axis. The best known classical inertia theorem is that of Sylvester : If P > 0 (positive definite) and H is Hermitian, then In PH=ln H. Less well known is Lyapunov's theorem [2 ] : There exists a P > 0 such that (R(AP) = i(AP+PA*)>0 if and only if lnA = (n, 0, 0). Both classical theorems are contained in a generalization (Taussky [4], Ostrowski-Schneider [3]) which we shall call the