Matrices whose hermitian part is positive definite

Abstract

We are concerned with the class ∏n of nxn complex matrices A for which the Hermitian part H(A) = A+A*/2 is positive definite. Various connections are established with other classes such as the stable, D-stable and dominant diagonal matrices. For instance it is proved that if there exist positive diagonal matrices D, E such that DAE is either row dominant or column dominant and has positive diagonal entries, then there is a positive diagonal F such that FA ϵ ∏n. Powers are investigated and it is found that the only matrices A for which Am ϵ ∏n for all integers m are the Hermitian elements of ∏n. Products and sums are considered and criteria are developed for AB to be in ∏n. Since ∏n n is closed under inversion, relations between H(A)-1 and H(A-1) are studied and a dichotomy observed between the real and complex cases. In the real case more can be said and the initial result is that for A ϵ ∏n, the difference H(adjA) - adjH(A) ≥ 0 always and is ˃ 0 if and only if S(A) = A-A*/2 has more than one pair of conjugate non-zero characteristic roots. This is refined to characterize real c for which cH(A-1) - H(A)-1 is positive definite. The cramped (characteristic roots on an arc of less than 180°) unitary matrices are linked to ∏n and characterized in several ways via products of the form A -1A*. Classical inequalities for Hermitian positive definite matrices are studied in ∏n and for Hadamard's inequality two types of generalizations are given. In the first a large subclass of ∏n in which the precise statement of Hadamardis inequality holds is isolated while in another large subclass its reverse is shown to hold. In the second Hadamard's inequality is weakened in such a way that it holds throughout ∏n. Both approaches contain the original Hadamard inequality as a special case.