Abstract
It is known that if A is positive definite Hermitian, then A· A -1⩾ I in the positive semidefinite ordering. Our principal new result is a converse to this inequality: under certain weak regularity assumptions about a function F on the positive definite matrices, A· F( A)⩾ AF( A) for all positive definite A if and only if F( A) is a positive multiple of A -1. In addition to the inequality A· A -1⩾ I, it is known that A· A -1 T ⩾ I and, stronger, that λ min ( A· B)⩾ λ min ( AB T ), for A, B positive definite Hermitian. We also show that λ min ( A· B)⩾ λ min ( AB) and note that λ min( AB) and λ min( AB T ) can be quite different for A, B positive definite Hermitian. We utilize a simple technique for dealing with the Hadamard product, which relates it to the conventional product and which allows us to give especially simple proofs of the closure of the positive definites under Hadamard multiplication and of the inequalities mentioned.
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