Mappings on matrices: invariance of functional values of matrix products

Abstract

AbstractLetMn, be the algebra of alln × nmatrices over a field F, wheren≧ 2. LetSbe a subset ofMncontaining all rank one matrices. We study mappings φ:S → Mn, such thatF(φ (A)φ (B)) =F(A B)for various families of functionsFincluding 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 allA= (aij) ∈Sfor some invertibleS ∈ Mn, field monomorphism σ of F, and an F*-valued mapping μ defined onS. 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 whenF:Mn→ {0, 1} is defined byF(X)= 0, ifX= 0, andF(X)= 1 otherwise. In such a case, one needs to characterize φ:S → Mnsuch that φ(A) φ (B) = 0 if and only ifAB= 0. We show that such a map has the standard form described above on rank one matrices inS.