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Mappings on matrices: invariance of functional values of matrix products

Part of: Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

Jor-Ting Chan ,
Chi-Kwong Li  and
Nung-Sing Sze
Jor-Ting Chan
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, e-mail: [email protected]
Chi-Kwong Li
Affiliation:
Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USA, e-mail: [email protected]
Nung-Sing Sze
Affiliation:
Department of Mathematics, The University of Hong Kong, Pokfulam Road, Hong Kong, and Department of Mathematics, University of Connecticut, Storrs, CT 06269-3009, USA, e-mail: [email protected]
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Abstract

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Let Mn, be the algebra of all n × n matrices over a field F, where n ≧ 2. Let S be a subset of Mn containing all rank one matrices. We study mappings φ: S → Mn, such that F(φ (A)φ (B)) = F(A B) for various families of functions F including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form A ↦ μ(A)S(σ (aij))S-1 for all A= (aij) ∈ S for some invertible S ∈ Mn, field monomorphism σ of F, and an F*-valued mapping μ defined on S. For real matrices, σ is often the identity map; for complex matrices, σ is often the identity map or the conjugation map: z ↦ z. A key idea in our study is reducing the problem to the special case when F:Mn → {0, 1} is defined by F(X) = 0, if X = 0, and F(X) = 1 otherwise. In such a case, one needs to characterize φ: S → Mn such that φ(A) φ (B) = 0 if and only if AB = 0. We show that such a map has the standard form described above on rank one matrices in S.

Keywords

zero product preserversunitary similarity invariant functions.

MSC classification

Secondary: 15A04: Linear transformations, semilinear transformations 15A60: Norms of matrices, numerical range, applications of functional analysis to matrix theory 15A18: Eigenvalues, singular values, and eigenvectors
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 165 - 184
Copyright
Copyright © Australian Mathematical Society 2006

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