Spectral decomposition theorem for real symmetric matrices in topoi and applications
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.
scientific article; zbMATH DE number 3906693