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Generalizations of a well-known result in matrix theory

Published online by Cambridge University Press:  18 May 2009

R. C. Thompson
R. C. Thompson
Affiliation:
The University of British Columbia, (Now at the University of California, Santa Barbara)
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Let A and C be m × m matrices and let B and D be n × n matrices, all with elements in a field F. Let AT denote the transpose of A. A well-known theorem states that, if every m × m matrix X for which AX = XA also satisfies CX = XC, then C = φ(A) for some polynomial φ(λ). In this note we establish the following simple generalizations.

Theorem 1. Let A and B have the same minimal polynomial m(λ). If each m × n matrix X over F for which AX = XB also satisfies CX = XD, then C = φ(A) and D = φ(B) for a polynomial φ(λ) over F.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 7 , Issue 1 , January 1965 , pp. 29 - 31
Copyright
Copyright © Glasgow Mathematical Journal Trust 1965

References

1.Perlis, S., The theory of matrices (Cambridge, Mass., 1952).Google Scholar