Inertia theorems for matrices: The semidefinite case

la. The inertia of an n Xn matrix A with complex elements is defined to be the integer triple In A = (7T(A), v(A), S(A», where 7T(A) {v(A)} is the number of eigenvalues of A in the open right {left} half-plane, and S(A) is the number of eigenvalues on the imaginary axis. The best-known classical theorem on inertias is that of Sylvester [I, I p. 296; 2], which may be stated as: If P > 0 (positive definite), and H is Hermitian, then In PH = In H. Lyapunov's theorem [3, p. 245; I, II p. 187; 4, 5] is less well-known: For a given A, there exists an H > 0 such that ~(AH) = t(AH + HA*) > 0 if and only if In A = (n, 0, 0).