## The distribution and moments of the smallest eigenvalue of a random matrix of Wishart type.(English)Zbl 0738.15010

Take an $$m\times n$$ $$(m\leq n)$$ random matrix $$X$$ in which each element is an independent standard normal random variable. Form the positive (semi) definite matrix $$A=XX^ T$$. The author shows how to obtain exact expressions for the distribution and the expected value of the smallest eigenvalue of $$A$$. The author gives new results giving the distribution as a simple recursion. This includes the more difficult case when $$n-m$$ is an even integer, without resorting to zonal polynomials and hypergeometric functions of matrix arguments. With the recursion, one can obtain exact expressions for the density and the moments of the distribution in terms of functions usually no more complicated than polynomials, exponentials, and at worst ordinary hypergeometric functions. The author further elaborates on the special cases when $$n- m=0,1,2$$, and 3 and gives a numerical table of the expected values for $$2\leq m\leq 25$$ and $$0\leq n-m\leq 25$$.
The paper contains the sections of introduction; main results; sample plots of the distributions; derivation of density formulas; expected values and other moments; $$n-m=0,1,2$$, and 3; computation of expected values; and appendices: mathematical programs; tables of expected values; sample formulas and other uses.
Reviewer: Y.Kuo (Knoxville)

### MSC:

 15B52 Random matrices (algebraic aspects) 15A42 Inequalities involving eigenvalues and eigenvectors 15A18 Eigenvalues, singular values, and eigenvectors

Mathematica
Full Text:

### References:

  Abramowitz, M.; Stegun, I. A., Handbook of Mathematical Functions (1970), Dover: Dover New York · Zbl 0515.33001  Anderson, T. W., An Introduction to Multivariate Statistical Analysis (1958), Wiley: Wiley New York · Zbl 0083.14601  Aomoto, K., Jacobi polynomials associated with Selberg integrals, SIAM J. Math. Anal., 18, 545-549 (1987) · Zbl 0639.33001  Edelman, A., Eigenvalues and condition numbers of random matrices, SIAM J. Matrix Anal. Appl., 543-560 (1988) · Zbl 0678.15019  Gradshteyn, I. S.; Ryzhik, I. W., Table of Integrals, Series, and Products (1965), Academic: Academic New York  James, A. T., Distributions of matrix variates and latent roots derived from normal samples, Ann. Math. Statist., 35, 475-501 (1964) · Zbl 0121.36605  Krishnaiah, P. R.; Cheng, T. C., On the exact distributions of the smallest roots of the Wishart matrix using zonal polynomials, Ann. Inst. Math. Statist., 23, 293-295 (1971) · Zbl 0276.62053  Muirhead, R. J., Aspects of Multivariate Statistical Theory (1982), Wiley: Wiley New York · Zbl 0556.62028  Silverstein, J. W., The smallest eigenvalue of a large-dimensional Wishart matrix, Ann. Probab., 13, 1364-1368 (1985) · Zbl 0591.60025  Spanier, J.; Oldham, K. B., An Atlas of Functions (1987), Hemisphere: Hemisphere Washington · Zbl 0618.65007  Wilks, S., Mathematical Statistics (1967), Wiley: Wiley New York · Zbl 0060.29502  Wolfram, S., Mathematica (1988), Addison-Wesley,: Addison-Wesley, Bedwood City, Calif.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.