On the Moments of Elementary Symmetric Functions of the Roots of Two Matrices

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.