Mappings on matrices: Invariance of functional values of matrix products

Abstract

Let M n be the algebra of all n × n matrices over a field double-struck F sign, where n ≥ 2. Let S be a subset of M n containing all rank one matrices. We study mappings Φ S → M n such that F(ø) (A)ø(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 → μ(A) S(σ(a ij))S -1 for all A = (a ij) ∈ S for some invertible S ∈ M n, field monomorphism σ of double-struck F sign*, and an double-struck F sign*-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 → Ž. A key idea in our study is reducing the problem to the special case when F : M n → {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 → M n 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. © 2006 Australian Mathematical Society.