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Linear Transformations on Algebras of Matrices

Published online by Cambridge University Press:  20 November 2018

Marvin Marcus  and
B. N. Moyls
Marvin Marcus
Affiliation:
University of British Columbia
B. N. Moyls
Affiliation:
University of British Columbia
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Let Mn 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 into Mn having one or more of the following properties:

  • (a) T(Rk) ⊆ for k = 1, …, n.

  • (b) 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.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 61 - 66
Copyright
Copyright © Canadian Mathematical Society 1959

References

The work of the first author was partially completed under U.SNational Science Foundation Grant No. NSF-G 5416. The work of the second author was supported in part by the United States Air Force Office of Scientific Research and Development Command.