Resumen de Mappings on matrices: invariance of functional values of matrix products

Jor-Ting Chan, Chi-Kwong Li, Nung-Sing Sze

• 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 $F$ 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 $F(X) = 0$, if $X = 0$, and $F(X)=1$ 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 $AB = 0$. We show that such a map has the standard form described above on rank one matrices in $\mathcal{S}$