Field of values and spectra of positive definite multiples

Abstract

the "field of values" of A. In [1, 2, 4] 1 the H -stable matrices were characterized. (A matrix A EM II (C) is called H -stable if AECT (HA) implies Re(A) > 0 for all H * = H > 0.) The simplest version of the characterization is that A is H-stable if and only if A is nonsingular and AEF (A) implies Re(A) > 0 or A = O. In this note we give a simple characterization of those AECT (HA) for some H * = H > 0 and the theorem on H -stability, for example, is an easy corollary. The characterization is constructive in that a specific class of positive definite matrices H for which AECT (H A) is produced. Let L == {H EM il (C) :H* = H > O} . We first make two observations: (I) OECT (HA) for some H E'i if and only if Ow' (KA) for all KE'i if and only if OECT (A);