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Characterizations of r-potent matrices

Published online by Cambridge University Press:  24 October 2008

Joseph P. McCloskey
Joseph P. McCloskey
Affiliation:
Department of Mathematics and Computer Science, University of Maryland Baltimore County, Catonville, MD 21228, U.S.A.
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A matrix A is said to be tripotent whenever A3 = A. The study of tripotent matrices is of statistical interest since if the n × 1 real random vector X follows an N(0, I) distribution and A is a symmetric matrix then the real quadratic form X′AX is distributed as the difference of two independently distributed X2 variates if and only if A3 = A. In fact, a necessary and sufficient condition that A is tripotent is that there exist two idempotent matrices B and C such that A = B – C, and BC = 0. Using properties of diagonalizable matrices, we will prove several algebraic characterizations of r-potent matrices that extend the known results for tripotent matrices. Our first result will be to obtain an analogous decomposition for an arbitrary r-potent matrix.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 96 , Issue 2 , September 1984 , pp. 213 - 222
Copyright
Copyright © Cambridge Philosophical Society 1984

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