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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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References

REFERENCES

[1]Anderson, T. W. and Styan, G. P. H.. Cochran’s Theorem, Rank Additivity, and Tripotent Matrices. Technical Report 43, Department of Statistics, Stanford University, (1980)Google Scholar
[2]Chipman, J. S. and Rao, M. M.. Projections, generalized inverses, and quadratic forms. J. Math. Anal. Appl. 9 (1964), 1–11.CrossRefGoogle Scholar
[3]Cochran, W. G.. The distribution of quadratic forms in a normal system with applications to the analysis of covariance. Proc. Cambridge Philos. Soc. 30 (1934), 178–181.CrossRefGoogle Scholar
[4]Drazin, M. P.. On diagonable and normal matrices. Quart. J. Math. 2 (1951), 189–198.CrossRefGoogle Scholar
[5]Graybill, F. A.. Matrices with Applications in Statistics (Wadsworth Publishing Company, 1983).Google Scholar
[6]Graybill, F. A. and Marsaglia, G.. Idempotent matrices and quadratic forms in the general linear hypothesis. Ann. Statist. 28 (1957), 676–686.Google Scholar
[7]Khatri, C. G.. Some results for the singular multivariate regression models. Sankhyä 30 (1968), 267–280.Google Scholar
[8]Perlis, S.. Theory of Matrices (Addison-Wesley, 1958).Google Scholar
[9]Pittenger, A. O., Personal communication, 1982.Google Scholar
[10]Rao, C. R. and Mitra, S. K.. Generalized Inverse of Matrices and Its Applications. (John Wiley, 1971).Google Scholar
[11]Styan, G. P. H.. Notes on the distribution of quadratic forms in singular normal variates. Biometrika 57 (1970), 567–572.CrossRefGoogle Scholar