Linear Transformations on Algebras of Matrices: The Invariance of the Elementary Symmetric Functions
- 1 January 1959
- journal article
- Published by Canadian Mathematical Society in Canadian Journal of Mathematics
- Vol. 11, 383-396
- https://doi.org/10.4153/cjm-1959-039-4
Abstract
In this paper we examine the structure of certain linear transformations T on the algebra of w-square matrices Mn into itself. In particular if A ∈ Mn 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 Er(T(A)) = Er(A) for A ∈ Mn 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 E3(T(A)) = E3(A) for all ∈ Mn is unknown to us.Keywords
This publication has 1 reference indexed in Scilit:
- Linear Transformations on Algebras of MatricesCanadian Journal of Mathematics, 1959