On Elementary Symmetric Functions of the Roots of Two Matrices in Multivariate Analysis

摘要

A lemma was proved to show that the moments of the (s-i)th elementary symmetric function (esf) in s non-null characteristic roots, lambda sub i (i = 1, 2, ..., s), of a matrix in multivariate analysis could be derived from those of the ith esf. Using this lemma the first four moments of the (s-1)th esf were obtained from those of the first esf already known (Pillai, 1954, 1960; Pillai and Samson, 1959). Further, a second lemma was given showing that the moments of the (s-1)th esf in the s characteristic roots, theta sub i = lambda sub i/(1 + lambda sub i), are derivable from those of the first esf in the lambda's. Upper percentage points (5% and 1%) were obtained for the distribution of the (s-1)th esf in the lambda's for s = 3 using the moment quotients. An example was given to illustrate the use of this criterion. (Author)