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Linear Transformations on Matrices: the Invariance of the Third Elementary Symmetric Function

Published online by Cambridge University Press:  20 November 2018

Leroy B. Beasley
Leroy B. Beasley*
Affiliation:
University of British Columbia, Vancouver, British Columbia
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Let T be a linear transformation on Mn the set of all n × n matrices over the field of complex numbers, . Let A ∈ Mn have eigenvalues λ1, …, λn and let Er(A) denote the rth elementary symmetric function of the eigenvalues of A :

Equivalently, Er(A) is the sum of all the principal r × r subdeterminants of A. T is said to preserve Er if Er[T(A)] = Er(A) for all A ∈ Mn. Marcus and Purves [3, Theorem 3.1] showed that for r ≧ 4, if T preserves Er then T is essentially a similarity transformation; that is, either T: A → UAV for all A ∈ Mn or T: A → UAtV for all A ∈ Mn, where UV = eiθIn, rθ ≡ 0 (mod 2π).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 22 , Issue 4 , 01 August 1970 , pp. 746 - 752
Copyright
Copyright © Canadian Mathematical Society 1970

References

1. Marcus, M. and Mine, H., A survey of matrix theory and matrix inequalities (Allyn and Bacon, Boston, 1964).Google Scholar
2. Marcus, M. and Moyls, B. N., Transformations on tensor product spaces, Pacific J. Math. 9 (1959), 1215–1221.Google Scholar
3. Marcus, M. and Purves, R., Linear transformations on algebras of matrices: The invariance of the elementary symmetric functions, Can. J. Math. 11 (1959), 383–396.Google Scholar
4. Marcus, M. and Westwick, R., Linear maps on skew-symmetric matrices: The invariance of the elementary symmetric functions, Pacific J. Math. 10 (1960), 917–924.Google Scholar