Two applications of the theory of primary matrix functions
type: Item
instance of: scholarly article
title: Two applications of the theory of primary matrix functions
zbMATH Open document ID: 1032.15018
DOI: 10.1016/S0024-3795(02)00267-7
author: Gregory G. Piepmeyer
author: Roger A. Horn
published in: Linear Algebra and its Applications
publication date: 2003-03-11T00:00:00Z
review text: A systematic exposition of primary matrix functions was given by \textit{R. A. Horn} and \textit{C. R. Johnson} [Topics in matrix analysis, Cambridge University Press, New York (1991; Zbl 0729.15001)]. For a complex \(n\times n\) matrix the authors use the Jordan canonical form \[ A=S\left[ \begin{matrix} J_{n_{1}}(\lambda) & 0 & 0 \\ 0 & \ddots & 0 \\ 0 & 0 & J_{n_{d}}(\lambda) \end{matrix} \right] S^{-1} \] where \(J_{k}(\lambda) \) are \(k\)-Jordan blocks. Assume that \(f:D\rightarrow\mathbb{C}\), \(D\subset\mathbb{C}\), is a given analytic function in an open set containing the spectrum of \(A.\) The primary matrix function is defined by \[ f\left( A\right) =S\left[ \begin{matrix} f(J_{n_{1}}(\lambda)) & 0 & 0\\ 0 & \ddots & 0\\ 0 & 0 & f(J_{n_{d}}(\lambda)) \end{matrix} \right] S^{-1} \] in which each block \(f( J_{k}( \lambda)) \) is an upper triangular Toeplitz matrix with the entries \(f( \lambda) ,\) \(f'( \lambda) ,\dots,f^{( k-1)}( \lambda) /( k-1)!\) in the first row. The authors consider whether \(f(A) \) can have rank one, and whether, for a given \(B,\) there is a unique \(A\) such that \(f(A) =B.\)
Mathematics Subject Classification ID: 15A54
Mathematics Subject Classification ID: 15A03
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 1880049
zbMATH Keywords: primary matrix function
zbMATH Keywords: matrix square root
zbMATH Keywords: matrix logarithm
zbMATH Keywords: tournament matrix
zbMATH Keywords: Jordan canonical form
zbMATH Keywords: Toeplitz matrix
zbMATH Keywords: rank
reviewed by: Jan Kubarski
MaRDI profile type: Publication
cites work: Combinatorial matrix theory
cites work: Q4040931
cites work: Uniqueness of matrix square roots and an application
full work available at URL: s0024-3795(02)00267-7
OpenAlex ID: W4206596179
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description: scientific article; zbMATH DE number 1880049