Symmetric matrices with respect to sesquilinear forms
type: Item
instance of: scholarly article
title: Symmetric matrices with respect to sesquilinear forms
zbMATH Open document ID: 1001.15019
DOI: 10.1016/S0024-3795(01)00598-5
author: Christian Mehl
author: Leiba Rodman
published in: Linear Algebra and its Applications
publication date: 2002-08-28T00:00:00Z
review text: Let \(V\) be a finite-dimensional vector space over the complex numbers \({\mathbb C}\) and let \([x,y]\) be a sesquilinear form on \(V.\) A linear transformation \(A\) will be called symmetric with respect to \([x,y]\) if \([Ax,y]=[x,Ay]\) for all \(x,y\in V.\) Let \(\langle\cdot,\cdot\rangle\) be the standard Euclidean inner product. Then \([x,y]=\langle Hx,y\rangle\) for some uniquely determined Hermitian matrix \(H\) and \(A\) is symmetric with respect to \(\langle Hx,y\rangle\) if \(HA=A^\star H.\) In this case, \(A\) is called \(H\)-selfadjoint. Canonical forms of \(H\)-selfadjoint matrices are well known for the case that \(H\) is invertible. The authors study the case where \(H\) is possibly singular. They show, if \(A\in{\mathbb C}^{n\times n}\) is an \(H\)-selfadjoint matrix, then there exists a nonsingular matrix \(P\in{\mathbb C}^{n\times n}\) such that \[ P^{-1}AP=A_{11}\oplus\cdots\oplus A_{kk} \text{and} P^\star HP=H_{11}\oplus\cdots\oplus H_{kk}, \] where for each \(j,\) the blocks \(A_{jj}\) and \(H_{jj}\) have the same sizes and are of one of three types described in the paper. The authors give an example to show that an indecomposable symmetric matrix with respect to a degenerate sesquilinear form may have arbitrarily many Jordan blocks.
Mathematics Subject Classification ID: 15A63
zbMATH DE Number: 1790242
zbMATH Keywords: degenerate inner product
zbMATH Keywords: sesquilinear form
zbMATH Keywords: selfadjoint matrix
zbMATH Keywords: linear transformation
zbMATH Keywords: canonical forms
zbMATH Keywords: indecomposable symmetric matrix
zbMATH Keywords: Jordan blocks
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1790242