On the matrix roots of \(f(X)=A\)
type: Item
instance of: scholarly article
title: On the matrix roots of \(f(X)=A\)
zbMATH Open document ID: 0656.15007
author: E. Spiegel
published in: Indian Journal of Pure \& Applied Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let A be an \(n\times n\) matrix with complex elements and let f(x) be a non-constant polynomial with complex coefficients. The author considers the problem of finding an \(n\times n\) matrix B with \(f(B)=A\). He shows, for example, that if A is the Jordan block with eigenvalue \(\lambda\), such a B exists if and only if either \(n=1\) or there exists a complex number \(\gamma\) such that \(f(\gamma)=\lambda\) and \(f'(\gamma)\neq 0\). In the case that A is an \(n\times n\) matrix with a single eigenvalue he gives necessary and sufficient conditions for the existence of an irreducible solution X to \(f(X)=A\). He also shows that if A is a nonderogatory matrix and I is the ideal generated by the characteristic polynomial of A then there exists a matrix B with \(f(B)=A\) if and only if there exists a polynomial g(x) with \(f(g(x))\equiv x(mod I)\).
Mathematics Subject Classification ID: 15A24
Mathematics Subject Classification ID: 15A21
zbMATH DE Number: 4071108
zbMATH Keywords: matrix roots
zbMATH Keywords: Jordan form
zbMATH Keywords: matrix equation
zbMATH Keywords: irreducible solution
zbMATH Keywords: nonderogatory matrix
zbMATH Keywords: characteristic polynomial
reviewed by: Fergus J. Gaines
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4071108