Fractional Hadamard powers of positive semidefinite matrices

Abstract

We consider the class S n of all real positive semidefinite n× n matrices, and the subclass S n + of all A∈ S n with non-negative entries. For a positive, non-integer number α and some A∈ S n + , when will the fractional Hadamard power A ♢ α again belong to S n + ? It is known that, for a specific α, this holds for all A∈ S n + if and only if α> n−2. Now let A∈ S n + be of the form A= T+ V, where T∈ S n + has rank 1 and V∈ S n has rank p⩾1. If the Hadamard quotient of T and V is Hadamard independent (‘in general position’) and V has ‘sufficently small’ entries, then a complete answer is given, depending on n, p, and α. Special attention is given to the case that p=1.