Orthogonal similarity and pairs of quadratic forms
type: Item
instance of: scholarly article
title: Orthogonal similarity and pairs of quadratic forms
zbMATH Open document ID: 0851.11022
DOI: 10.1016/0024-3795(95)90018-7
author: William C. Waterhouse
published in: Linear Algebra and its Applications
publication date: 1996-11-25T00:00:00Z
review text: Let \(L/K\) be a finite separable field extension and let \(W\) be a \(K\)-space with an endomorphism \(S\) such that \(L = K[S]\). For any \(L\)-bilinear form \(C\) on \(W\) the trace map \(\text{Tr} = \text{tr}_{L/K}\) provides us with a \(K\)-bilinear form \(B = \text{Tr} \circ C\) such that \(B(Sx, y) = B(x,Sy)\) for \(x,y \in W\). In the paper the author examines the inverse process, that is, he shows how to get (up to isometry) the form \(C\) when the form \(\text{Tr} \circ C\) is known. As a result he gets that the two known ways of classifying rational symmetric matrices under orthogonal similarity, first presented by \textit{S. Friedland} [Linear Algebra Appl. 192, 109-114 (1993; Zbl 0787.15011)] and the second by the author [Invent. Math. 37, 157-164 (1976; Zbl 0337.10015)], are in fact the same.
Mathematics Subject Classification ID: 11E12
Mathematics Subject Classification ID: 15A21
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 841732
zbMATH Keywords: symmetric bilinear form
zbMATH Keywords: trace form
zbMATH Keywords: pairs of quadratic forms
zbMATH Keywords: orthogonal similarity
zbMATH Keywords: separable field extension
zbMATH Keywords: rational symmetric matrices
reviewed by: Andrzej Sładek
MaRDI profile type: Publication
full work available at URL: 0024-3795(95)90018-7
OpenAlex ID: W2038354885
cites work: Q3344268
cites work: Rational orthogonal similarity of rational symmetric matrices
cites work: Pairs of quadratic forms
cites work: Self-adjoint operators and formally real fields
cites work: Scaled trace forms over number fields
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description: scientific article; zbMATH DE number 841732