# zbMATH — the first resource for mathematics

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:
 [1] Abramowitz, M.; Stegun, I.A., Handbook of mathematical functions, (1970), Dover New York · Zbl 0515.33001 [2] Anderson, T.W., An introduction to multivariate statistical analysis, (1958), Wiley New York · Zbl 0083.14601 [3] Aomoto, K., Jacobi polynomials associated with Selberg integrals, SIAM J. math. anal., 18, 545-549, (1987) · Zbl 0639.33001 [4] Edelman, A., Eigenvalues and condition numbers of random matrices, SIAM J. matrix anal. appl., 543-560, (1988) · Zbl 0678.15019 [5] Gradshteyn, I.S.; Ryzhik, I.W., Table of integrals, series, and products, (1965), Academic New York [6] James, A.T., Distributions of matrix variates and latent roots derived from normal samples, Ann. math. statist., 35, 475-501, (1964) · Zbl 0121.36605 [7] 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 [8] Muirhead, R.J., Aspects of multivariate statistical theory, (1982), Wiley New York · Zbl 0556.62028 [9] Silverstein, J.W., The smallest eigenvalue of a large-dimensional Wishart matrix, Ann. probab., 13, 1364-1368, (1985) · Zbl 0591.60025 [10] Spanier, J.; Oldham, K.B., An atlas of functions, (1987), Hemisphere Washington · Zbl 0618.65007 [11] Wilks, S., Mathematical statistics, (1967), Wiley New York · Zbl 0060.29502 [12] Wolfram, S., Mathematica, (1988), 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.