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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
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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

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