On rational invariants of a set of symmetric matrices
type: Item
instance of: scholarly article
title: On rational invariants of a set of symmetric matrices
zbMATH Open document ID: 0532.15021
DOI: 10.1016/0024-3795(83)90203-3
author: Boris Raykhshteyn
published in: Linear Algebra and its Applications
publication date: 1983-01-01T00:00:00Z
review text: A polynomial f is called an invariant of an operator A:\(L\to L\), with L a vector n-space, if for any regular linear operator D:\(L\to L\) one has \(f(A)=f(DAD^{-1})\). The ring of invariants of A is generated by the coefficients of the characteristic polynomial of A. The straightforward generalizations of this statement to the case of two operators (A,B) is not true. However, starting with some identities resulting from the Cayley-Hamilton theorem the author obtains the following (a) for \(k=1,2,...,n-1\) a condition is found for a pair (A,B) of symmetric operators acting in Euclidean n-space to have common invariant k-subspace (provided that A does not have multiple eigenvalues); (b) it is shown that the field of rational invariants of (A,B) is isomorphic to a subfield of a rational function field with \(n(n+3)/2\) generators consisting of elements symmetric with respect to the permutation group \(P_ n\); (c) it is shown that any rational invariant of \((g+2)\) symmetric operators \(A,B,C_ 1,C_ 2,...,C_ g\) can be expressed as a rational function of invariants of one or two operators that are taken for pairs (A,B), \((A,C_ 1)\), \((A,C_ 2),...,(A,C_ g)\), \((A,B+C_ 1)\), \((A,B+C_ 2),...,(A,B+C_ g)\).
Mathematics Subject Classification ID: 15A72
Mathematics Subject Classification ID: 15A15
zbMATH DE Number: 3843933
zbMATH Keywords: ring of invariants
zbMATH Keywords: characteristic polynomial
zbMATH Keywords: Cayley-Hamilton theorem
zbMATH Keywords: symmetric operators
zbMATH Keywords: invariant k-subspace
zbMATH Keywords: rational invariants
reviewed by: J. de Graaf
MaRDI profile type: Publication
full work available at URL: 0024-3795(83)90203-3
OpenAlex ID: W2022964221
cites work: Q5639811
cites work: The invariant theory of \(n\times n\) matrices
cites work: The center of the ring of 3×3 generic matrices
cites work: Invariant rational functions and a problem of Steenrod
description: scientific article; zbMATH DE number 3843933