Linear Transformations on Algebras of Matrices

TLDR

The paper studies the algebra \(M_n\) of \(n\times n\) complex matrices, along with the unimodular group \(U_n\), Hermitian matrices \(H_n\), rank‑\(k\) matrices \(R_k\), and the eigenvalue set \(\operatorname{ev}(A)\). It aims to characterize the structure of linear maps \(T:M_n\to M_n\) that preserve rank subsets, the unimodular group, determinants on Hermitian matrices, and eigenvalues on Hermitian matrices. Such maps are linear (\(T(aA+bB)=aT(A)+bT(B)\)) but not assumed multiplicative, and are required to satisfy properties (a)–(d) listed above.

Abstract

Let M n denote the algebra of n-square matrices over the complex numbers; and let U n , H n , and R k denote respectively the unimodular group, the set of Hermitian matrices, and the set of matrices of rank k , in M n . Let ev(A) be the set of n eigenvalues of A counting multiplicities. We consider the problem of determining the structure of any linear transformation (l.t.) T of M n into M n having one or more of the following properties: (a) T(R k ) ⊆ for k = 1, …, n. (b) T(U n ) ⊆ Un (c) det T(A) = det A for all A ∈ H n . (d) ev(T(A)) = ev(A) for all A ∈ H n . We remark that we are not in general assuming that T is a multiplicative homomorphism; more precisely, T is a mapping of Mn into itself, satisfying T(aA + bB) = aT(A) + bT(B) for all A, B in Mn and all complex numbers a, b .