Some Theorems on the Inertia of General Matrices

1.1. Much is known about the distribution of the roots of algebraic equations in half-planes. (Cf. the corresponding parts in the survey [l] by Marden.) In the case of matrix equations, however, there appears to be only one known general result concerning the location of the eigenvalues of a matrix in the left half-plane. This theorem is generally known as Lyapunov’s theorem: (L,) Let A be an n-th order matrix with complex elements, and let C be an n-th order positive dejkite Hermitian matrix. Then there exists a negative definite matrix H for which AH+HA*=C (1) holds, if and only $a11 eigenvalues of A have negative real part. The real case of this theorem is a special case of some theorems proved by Lyapunov [2, p. 276-2771, establishing conditions for the stability of solutions of differential equations. Bellman [3, p. 245; cf. also 141 and Gantmacher [4, Vol. II, p. 1891 give proofs of this theorem, which make use of differential equations. An algebraic proof has been given by Hahn [5]. 1.2. Recently, investigations into the behavior of economic systems depending on a finite number of parameters have led Arrow and McManus [6] to