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On the matrix equation (AX)n = ∥A∥I

Published online by Cambridge University Press:  24 October 2008

M. E. Grimshaw
M. E. Grimshaw
Affiliation:
Newnham CollegeCambridge
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Extract

It was proved by J. Williamson (1) that when C is any circulant matrix of order n and Ω is a certain diagonal matrix then

where ∥ C ∥ is the determinant of C and I is the unit matrix of order n. A simpler proof was given by U. Wegner (2), and later the characteristic equation of ΩC was discussed independently by L. Toscano (3). In the present note we prove the more general theorem that there corresponds to any matrix A of order n with simple elementary divisors a solution X of the equation (AX)n = ∥ A ∥ I. The theorem has a simple geometrical interpretation; the proof is almost immediate.

Type
Research Notes
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 44 , Issue 2 , April 1948 , pp. 292 - 294
Copyright
Copyright © Cambridge Philosophical Society 1948

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References

REFERENCES

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