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
having one or more of the following properties:
T(Rk) ⊆ for k = 1, …, n.
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.