Rang des matrices hermitiennes semi-definies positives indefiniment divisibles
type: Item
instance of: scholarly article
title: Rang des matrices hermitiennes semi-definies positives indefiniment divisibles
zbMATH Open document ID: 0358.15019
DOI: 10.1016/0024-3795(76)90024-0
author: Paulette Farjot
published in: Linear Algebra and its Applications
publication date: 1976-01-01T00:00:00Z
review text: In this paper we study \(n \times n\) hermitian semi definite positive matrices \(A= (a(i,j))\) which are infinitely divisible in \textit{R. A. Horn's} sense [Trans. Am. Math. Soc. 136, 269--286 (1969; Zbl 0177.05003)], i.e. such that it is possible to find an \(n\times n\) hermitian matrix \(U = (u(i,j))\) satisfying: i) \(a (i,j) = e^{u (i,j)}\) if \(a (i,j) \neq 0\); ii) for all \(\theta > 0\), \(A_\theta = (a_\theta (i,j))\) is hermitian semi definite positive where \(A_\theta\) is defined by \(a_\theta (i,j) = e^{\theta u (i,j)}\) if \(a (i,j) \neq 0\); \(a_\theta (i, j) =0\) if \(a (i, j) =0\). The main result we prove is: ``All those matrices \(A_\theta\) (\(\theta >0\)), called the ``Hadamard power matrices of \(A\)'' have the same rank''. In addition, we give a representation theorem for matrices which are infinitely divisible in \(\mathbb R\) (such that the elements \(a_\theta (i, j)\) can be chosen real).
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 3557960
MaRDI profile type: Publication
full work available at URL: 0024-3795(76)90024-0
OpenAlex ID: W2013116677
cites work: Q4405419
cites work: Q5508985
cites work: The Theory of Infinitely Divisible Matrices and Kernels
cites work: Positive definite kernels, continuous tensor products, and central limit theorems of probability theory
cites work: Hadamard products and multivariate statistical analysis
description: scientific article; zbMATH DE number 3557960