Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-24T00:48:53.180Z Has data issue: false hasContentIssue false
  • English
  • Français

A lower bound for the degree of polynomials satisfied by matrices

Part of: Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

K. R. Pearson
K. R. Pearson
Affiliation:
La Trobe UniversityBundoora, Victoria, 3083 Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

R. Paré and W. Schelter (1978) have extended the Cayley-Hamilton theorem by showing that for each n<1 there is an integer k such that all n x n matrices over any (possibly noncommutative) ring satisfy a monic polynomial of degree k. We give a lower bound for this degree, namely π(n), which is defined as the shortest possible length of a sequence with entries from {1, 2, …, n}.

Keywords

matrixnoncommutative ringpolynomialCayley-Hamilton theorem

MSC classification

Secondary: 15A24: Matrix equations and identities
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 4 , June 1979 , pp. 430 - 436
Copyright
Copyright © Australian Mathematical Society 1979

References

Adleman, L. (1974), ‘Short permutation strings’, Discrete Math. 10, 197–200.CrossRefGoogle Scholar
Harary, Frank, Norman, Robert Z. and Cartwright, Dorwin (1965), Structural models: an introduction to the theory of directed graphs (John Wiley, New York).Google Scholar
Kleitman, D. J. and Kwiatkowski, D. J. (1976), ‘A lower bound on the length of a sequence containing all permutations as subsequences’, J. Comb. Theory 21, 129–136.CrossRefGoogle Scholar
Koutas, P. J. and Hu, T. C. (1975), ‘Shortest string containing all permutations’, Discrete Math. 11, 125–132.CrossRefGoogle Scholar
Paré, R. and Schelter, W. (1978), ‘Finite extensions are integral’, J. Alg. 53, 477–479.CrossRefGoogle Scholar
Robson, J. C. (1979), ‘Polynomials satisfied by matrices’, J. Alg. 55, 509–520.CrossRefGoogle Scholar
Wilson, Robin J. (1972), Introduction to graph theory (Oliver and Boyd, Edinburgh).Google Scholar