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Matrices with Prescribed Characteristic Polynomials

Published online by Cambridge University Press:  20 January 2009

H. K. Farahat  and
W. Ledermann
H. K. Farahat
Affiliation:
The University of Sheffield
W. Ledermann
Affiliation:
The University of Manchester
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It is well known that every monic polynomial of degree n with coefficients in a field Φ is the characteristic polynomial of some n × n matrix A with elements in in Φ . However, it is clear that this result is an extremely weak one, and that it should be possible to impose considerable restrictions upon the matrix A. In this note we prove two results in this direction. In section 2, we show that it is possible to prescribe all but one of the diagonal elements of A. This result was first proved by Mirsky (2) when the ground field Φ is the field of complex numbers. In section 3, we see that we can require A to have any prescribed non-derogatory n–l × n–1 matrix in the top left-hand corner.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 11 , Issue 3 , May 1959 , pp. 143 - 146
Copyright
Copyright © Edinburgh Mathematical Society 1959

References

REFERENCES

Jacobson, N., Lectures on Abstract Algebra, II, p. 69.Google Scholar
Mirsky, L., Matrices with prescribed characteristic roots and diagonal elements, J. London Math. Soc., 33, 1 (1958), No. 129, pp. 14–21.CrossRefGoogle Scholar