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December, 1964 On the Moments of Elementary Symmetric Functions of the Roots of Two Matrices
K. C. Sreedharan Pillai
Ann. Math. Statist. 35(4): 1704-1712 (December, 1964). DOI: 10.1214/aoms/1177700392

Abstract

A lemma is given first which provides an easy method of expressing the product of an $s$th order Vandermonde type determinant and the $k$th and $l$th $(k, l \geqq 0)$ powers of the $r$th and $h$th $(r, h \leqq s)$ elementary symmetric functions (esf's) respectively as a linear compound of determinants. The lemma extends itself readily to the product of powers of any number of esf's up to the $s$th. Using this lemma and some reduction formulae for certain special types of Vandermonde type determinants, a second lemma has been proved to show that certain formulas for the moments of esf's in $s$ non-null characteristic roots $\lambda_i(0 < \lambda_1 \leqq \lambda_2 \leqq \cdots \leqq \lambda_s < \infty)$ of a matrix can be easily derived from corresponding formulas for the moments of corresponding esf's in $s$ non-null roots, $\theta_i, (0 < \theta_1 \leqq \cdots \leqq \theta_s < 1)$ of another matrix and vice versa. Illustrations are given explaining both lemmas.

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K. C. Sreedharan Pillai. "On the Moments of Elementary Symmetric Functions of the Roots of Two Matrices." Ann. Math. Statist. 35 (4) 1704 - 1712, December, 1964. https://doi.org/10.1214/aoms/1177700392

Information

Published: December, 1964
First available in Project Euclid: 27 April 2007

zbMATH: 0129.11605
MathSciNet: MR168062
Digital Object Identifier: 10.1214/aoms/1177700392

Rights: Copyright © 1964 Institute of Mathematical Statistics

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Vol.35 • No. 4 • December, 1964
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