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

Published online by Cambridge University Press:  20 November 2018

Marvin Marcus  and
Roger Purves
Marvin Marcus
Affiliation:
University of British Columbia
Roger Purves
Affiliation:
University of British Columbia
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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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 383 - 396
Copyright
Copyright © Canadian Mathematical Society 1959

References

1. Marcus, M. and Moyls, B. N., Linear transformations on algebras of matrices, Can. J. Math., 11 (1959), 61-6.Google Scholar