Linear Transformations on Algebras of Matrices
Marvin D. Marcus, B. N. Moyls · Canadian Journal of Mathematics · 1959
Let Mn denote the algebra of n-square matrices over the complex numbers; and let Un, Hn, and Rk denote respectively the unimodular group, the set of Hermitian matrices, and the set of matrices of rank k, in Mn. 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 Mn into Mn having one or more of the following properties: (a) T(Rk) ⊆ for k = 1, …, n. (b) T(Un) ⊆ Un (c) det T(A) = det A for all A ∈ Hn. (d) ev(T(A)) = ev(A) for all A ∈ Hn. 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.