Mappings on matrices: invariance of functional values of matrix products

J. Aust. Math. Soc. 81 (2006), 165-184

Mappings on matrices: invariance of functional values of matrix products

Jor-Ting Chan
Department of Mathematics
The University of Hong Kong
Pokfulam Road
Hong Kong
jtchan@hku.hk

Chi-Kwong Li
Department of Mathematics
College of William and Mary
Williamsburg, VA 23187--8795
USA
ckli@math.wm.edu

and

Nung-Sing Sze
Department of Mathematics
The University of Hong Kong
Pokfulam Road
Hong Kong
and
Department of Mathematics
University of Connecticut
Storrs, CT 06269-3009
USA
sze@math.uconn.edu

Abstract

Let $\mathcal{M}_n$ be the algebra of all $n\times n$ matrices over a field $\mathbb{F}$ , where $n \ge 2$ . Let $\mathcal{S}$ be a subset of $\mathcal{M}_n$ containing all rank one matrices. We study mappings $\phi: \mathcal{S} \rightarrow \mathcal{M}_n$ such that $F(\phi(A)\phi(B)) = F(AB)$ for various families of functions

including all the unitary similarity invariant functions on real or complex matrices. Very often, these mappings have the form $A \mapsto \mu(A) S(\sigma(a_{ij}))S^{-1}$ for all $A = (a_{ij}) \in \mathcal{S}$ for some invertible $S\in \mathcal{M}_n$ , field monomorphism $\sigma$ of $\mathbb{F}$ , and an $\mathbb{F}^*$ -valued mapping $\mu$ defined on $\mathcal{S}$ . For real matrices, $\sigma$ is often the identity map; for complex matrices, $\sigma$ is often the identity map or the conjugation map: $z \mapsto \bar z$ . A key idea in our study is reducing the problem to the special case when $F: \mathcal{M}_n \to \{0, 1\}$ is defined by

, if

, and

otherwise. In such a case, one needs to characterize $\phi: \mathcal{S} \rightarrow \mathcal{M}_n$ such that $\phi(A)\phi(B) = 0$ if and only if

. We show that such a map has the standard form described above on rank one matrices in $\mathcal{S}$ .

Download the article in PDF format (size 163 Kb)

Australian Mathematical Publishing Association Inc.