The spectral mapping theorem, norms on rings, and resultants.
type: Item
instance of: scholarly article
title: The spectral mapping theorem, norms on rings, and resultants.
zbMATH Open document ID: 1085.15504
author: V. Pereyra
published in: L'Enseignement Mathématique. 2e Série
publication date: 2000-01-01T00:00:00Z
review text: From the paper: It is a well-known and useful result in spectral theory of complex finite dimensional vector spaces that if the characteristic polynomial of an \(n\times n\)-matrix \(M\) splits as \(P_M (t)=\det(tI_n-M)=\prod^n_{i=1}(t-\lambda_i)\) then, for any polynomial \(F(x)\), we have that \(\det(tI_n-F(M))=\prod^n_{i=1}(t-F(\lambda_i))\). We call this result the Spectral Mapping Theorem, because it is similar to the Spectral Mapping Theorem for Banach algebras. We give here a short, simple and self-contained proof of the Spectral Mapping Theorem for matrices with entries in an arbitrary commutative ring. The result is placed in the wider framework of norms on algebras. It is shown that the Spectral Mapping Theorem follows from a uniqueness result for norms on polynomial rings in one variable. The results are used to generalize classical formulas for the resultant of polynomials.
Mathematics Subject Classification ID: 15A60
Mathematics Subject Classification ID: 15A18
Mathematics Subject Classification ID: 47A60
zbMATH DE Number: 1736564
zbMATH Keywords: characteristic polynomial
zbMATH Keywords: Spectral Mapping Theorem
zbMATH Keywords: norms on polynomial rings
zbMATH Keywords: resultant of polynomials
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1736564