Fractional Hadamard powers of positive semidefinite matrices.
The authors consider the class \(\varphi_n\) of all real positive semidefinite \(n\times n\) matrices, and the subclass \(\varphi^+_n\) of all \(A\in\varphi_n\) with non-negative entries. For a positive, non-integer number \(\alpha\) and some \(A\in \varphi_n^+\), when will the fractional Hadamard power \(A^{\diamondsuit \alpha}\) again belong to \(\varphi_n^+\)? It is known that, for a specific \(\alpha\), this holds for all \(A\in\varphi^+_n\) if and only if \(\alpha>n-2\). Now let \(A\in \varphi_n^+\) be the form \(A=T+V\), where \(T\in\varphi_n^+\) has rank 1 and \(V\in\varphi_n\) has rank \(p\geq 1\). If the Hadamard quotient of \(T\) and \(V\) is Hadamard independent and \(V\) has sufficiently small entries, then a complete answer is given, depending on \(n\), \(p\), and \(\alpha\). Special attention is given to the case that \(p=1\). The paper contains the sections Hadamard powers; Positive semi-definite matrices; Clouds; Polynomial coefficients; The Hadamard span; A Taylor expansion; The main theorem; The case of lowest rank; Sign patterns.
scientific article; zbMATH DE number 1978106
scientific article