Matricial inner products and pointed cones of Hermitian-preserving linear transformations
type: Item
instance of: scholarly article
title: Matricial inner products and pointed cones of Hermitian-preserving linear transformations
zbMATH Open document ID: 0871.15030
DOI: 10.1016/S0024-3795(96)00205-4
author: Joseph R. Siler
author: Richard D. Hill
published in: Linear Algebra and its Applications
publication date: 1997-09-28T00:00:00Z
review text: Let \(V\) be a vector space over the real or complex numbers. A subset \(K\) of \(V\) is called a cone if for \(x,y\) in \(K\) and \(\alpha \geq 0\) the vector \(\alpha x+y\) is also in \(K\). Let \(M_n\) be the set of \(n\times n\) matrices with real or complex entries. The authors introduce a matricial inner product as a function \(M_n \times M_n \to M_n\), \((A,B) \to B^*PA+ AQB^*\), where \(P\) and \(Q\) are positive semidefinite and at least one is positive definite. They show that this function is a vectorial inner product with respect to the reflector \(\sigma\), where \(\sigma (A) =A^*\). The matricial inner product is used to show that certain sets of matrices are cones.
Mathematics Subject Classification ID: 15A63
Mathematics Subject Classification ID: 15A04
Mathematics Subject Classification ID: 15B48
zbMATH DE Number: 1011323
zbMATH Keywords: vector space
zbMATH Keywords: cone
zbMATH Keywords: matricial inner product
zbMATH Keywords: vectorial inner product
reviewed by: Erich W. Ellers
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1011323