How Many Matrices Have Roots?

Abstract

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.