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How Many Matrices Have Roots?

  • J. M. Borwein (a1) and B. Richmond (a2)
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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.

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References
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1. Andrews, G. E., The theory of partitions (Addison-Wesley, 1976).
2. Chandrasekharan, K., Arithmetical functions (Springer-Verlag, 1970).
3. Gantmacher, F. R., Matrix theory, Vol. 1 (Chelsea, 1959).
4. Kreis, H., Auflösung der Gleichung Xn = A, Vischr. natuforsch. Ges. Zurich 53 (1908), 366376.
5. MacDuffee, C. C., The theory of matrices (Chelsea Publishing Company, New York, 1946).
6. Meinardus, G., Asymptotische Aussogen über Partitionen, Math. Z. 59 (1954), 388398.
7. Pedis, S., Theory of matrices, 3rd Edition (Addison-Wesley, 1958).
8. van der Waerden, B. L., Modem algebra, Vols. I and II (Frederick Ungar Co., 1949 and 1950).
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Canadian Journal of Mathematics
  • ISSN: 0008-414X
  • EISSN: 1496-4279
  • URL: /core/journals/canadian-journal-of-mathematics
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