Mappings on matrices: Invariance of functional values of matrix products

Let Mnbe the algebra of all n × n matrices over a field double-struck F sign, where n ≥ 2. Let S be a subset of Mncontaining all rank one matrices. We study mappings Φ S → Mnsuch 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(aij))S-1for all A = (aij) ∈ S for some invertible S ∈ Mn, 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 : 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 → Mnsuch 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.