Fractional Hadamard powers of positive semidefinite matrices.
type: Item
instance of: scholarly article
title: Fractional Hadamard powers of positive semidefinite matrices.
zbMATH Open document ID: 1041.15012
DOI: 10.1016/S0024-3795(03)00421-X
author: D. Massart
published in: Linear Algebra and its Applications
publication date: 2003-09-09T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 15A30
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1978106
zbMATH Keywords: Positive semidefinite matrix
zbMATH Keywords: Gram matrix
zbMATH Keywords: Hadamard product
zbMATH Keywords: fractional Hadamard power
zbMATH Keywords: Hadamard independent matrix
zbMATH Keywords: Hadamard span
zbMATH Keywords: cloud
zbMATH Keywords: quasi linearly independent set
zbMATH Keywords: multiplicative trace
reviewed by: Yueh-er Kuo
MaRDI profile type: Publication
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full work available at URL: s0024-3795(03)00421-x
OpenAlex ID: W2045862997
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description: scientific article; zbMATH DE number 1978106