Spectral decomposition theorem for real symmetric matrices in topoi and applications
type: Item
instance of: scholarly article
title: Spectral decomposition theorem for real symmetric matrices in topoi and applications
zbMATH Open document ID: 0568.18004
DOI: 10.1016/0022-4049(85)90058-1
author: Christiane Rousseau
published in: Journal of Pure and Applied Algebra
publication date: 1985-01-01T00:00:00Z
review text: The connection between the constructive arguments in a topos and arguments involving parameters was presented for the first time by the author in 'Topos theory and complex analysis' [Lect. Notes Math. 753, 623-659 (1979; Zbl 0433.32003)]. In this paper she gives the constructive version (in the topos-theoretic sense) of the spectral decomposition theorem for real symmetric matrices. By interpretating this result in spatial topoi the author obtains: a normal form for real symmetric matrices depending continuously (resp. differentiably) on parameters; a versal deformation of any real symmetric matrix; and the following result: if a real symmetric matrix depends continuously on parameters, then its eigenvalues depend continuously on the same parameters. In the proofs orthogonal matrices can be replaced by unitary matrices obtaining similar theorems for complex Hermitian matrices.
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 15B57
Mathematics Subject Classification ID: 15A99
zbMATH DE Number: 3906693
zbMATH Keywords: spectral decomposition theorem
zbMATH Keywords: real symmetric matrices
zbMATH Keywords: spatial topoi
zbMATH Keywords: normal form
zbMATH Keywords: parameters
zbMATH Keywords: versal deformation
zbMATH Keywords: eigenvalues
reviewed by: I. Tofan
MaRDI profile type: Publication
full work available at URL: 0022-4049(85)90058-1
OpenAlex ID: W2033766694
cites work: ON MATRICES DEPENDING ON PARAMETERS
cites work: Q5182330
cites work: Topos theory and complex analysis
cites work: Q4198913
cites work: Q4477898
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description: scientific article; zbMATH DE number 3906693