Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-20T12:21:53.707Z Has data issue: false hasContentIssue false
  • English
  • Français

How Many Matrices Have Roots?

Published online by Cambridge University Press:  20 November 2018

J. M. Borwein  and
B. Richmond
J. M. Borwein
Affiliation:
Dalhousie University, Halifax, Nova Scotia
B. Richmond
Affiliation:
University of Waterloo, Waterloo, Ontario
Rights & Permissions [Opens in a new window]

Extract

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.

In many basic linear algebra texts it is shown that various classes of square matrices (normal, positive, invertible) possess square roots. In this note we characterize those n × n matrices with complex entries which possess at least one square root without any restriction on the class of root or matrix involved. We then use this characterization to obtain asymptotic estimates for the relative profusion of such matrices.

In Section 1 we characterize those n × n matrices with entries in C (or any algebraically complete field) which have square roots over C. This characterization is in terms of similarity classes. In Section 2 we give asymptotic estimates for the number of Jordan forms of nilpotent n × n matrices which are squares. Section 3 is given over to numerical results concerning the actual and asymptotic frequency of such forms.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 36 , Issue 2 , 01 April 1984 , pp. 286 - 299
Copyright
Copyright © Canadian Mathematical Society 1984

References

1. Andrews, G. E., The theory of partitions (Addison-Wesley, 1976).Google Scholar
2. Chandrasekharan, K., Arithmetical functions (Springer-Verlag, 1970).CrossRefGoogle Scholar
3. Gantmacher, F. R., Matrix theory, Vol. 1 (Chelsea, 1959).Google Scholar
4. Kreis, H., Auflösung der Gleichung Xn = A, Vischr. natuforsch. Ges. Zurich 53 (1908), 366–376.Google Scholar
5. MacDuffee, C. C., The theory of matrices (Chelsea Publishing Company, New York, 1946).Google Scholar
6. Meinardus, G., Asymptotische Aussogen über Partitionen, Math. Z. 59 (1954), 388–398.Google Scholar
7. Pedis, S., Theory of matrices, 3rd Edition (Addison-Wesley, 1958).Google Scholar
8. van der Waerden, B. L., Modem algebra, Vols. I and II (Frederick Ungar Co., 1949 and 1950).Google Scholar