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Matrix Commutators

Published online by Cambridge University Press:  20 November 2018

M. F. Smiley
M. F. Smiley*
Affiliation:
University of Iowa and University of California, Riverside
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A classical theorem states that if a square matrix B over an algebraically closed field F commutes with all matrices X over F which commute with a matrix A over F, then B must be a polynomial in A with coefficients in F (2). Recently Marcus and Khan (1) generalized this theorem to double commutators. Our purpose is to complete the generalization to commutators of any order.

Let F be an algebraically closed field and let Fn be the ring of all n by n matrices with elements in F. We define ΔYZ — = [Z, Y] = ZY — YZ for all Y, Z in Fn.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 353 - 355
Copyright
Copyright © Canadian Mathematical Society 1961

References

1. Marcus, M. and Khan, N. A., On matrix commutators, Can. J. Math., 12 (1960), 269–277.Google Scholar
2. Wedderburn, J. H. M., Lectures on matrices, Amer. Math. Soc. Colloq. Pub., 17 (New York, 1934).Google Scholar