Linear Transformations on Algebras of Matrices: The Invariance of the Elementary Symmetric Functions

Abstract

In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices M n into itself. In particular if A ∈ M n let Er(A) be the rth elementary symmetric function of the eigenvalues of A . Our main result states that if 4 ≤ r ≤ n — 1 and E r (T(A)) = Er(A) for A ∈ M n then T is essentially (modulo taking the transpose and multiplying by a constant) a similarity transformation: No such result as this is true for r = 1,2 and we shall exhibit certain classes of counterexamples. These counterexamples fail to work for r = 3 and the structure of those T such that E 3 (T(A)) = E 3 (A) for all ∈ M n is unknown to us.