On symmetrizing a matrix
type: Item
instance of: scholarly article
title: On symmetrizing a matrix
zbMATH Open document ID: 0648.65038
author: V. Ch. Venkaiah
author: Syamal K. Sen
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1988-01-01T00:00:00Z
review text: The paper presents an algorithm to compute a matrix symmetrizer. A symmetrizer of an arbitrary square matrix A is a symmetric matrix X satisfying the matrix equation \(XA=A^ tX\), where t indicates the transpose. It is shown that a symmetrizer reduces a nonsymmetric real eigenvalue problem into a symmetric one, which is relatively easier to solve. Moreover, it is shown that a symmetrizer can be used to obtain the zeros of a real polynomial. It is also useful in stability problems of control theory. The paper presents an algorithm to compute a symmetrizer, which requires a finite number of arithmetic operations, superimposed on a method of solving linear equations. This algorithm includes as a special case a recursive algorithm previously known for lower Hessenberg matrices. The stability of the algorithm and an example are discussed.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 4058750
zbMATH Keywords: matrix symmetrizer
zbMATH Keywords: matrix equation
zbMATH Keywords: nonsymmetric real eigenvalue problem
zbMATH Keywords: zeros of a real polynomial
zbMATH Keywords: stability problems of control theory
zbMATH Keywords: lower Hessenberg matrices
reviewed by: Franco Flandoli
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4058750