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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.

Information

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.