On the Distribution of the Latent Vectors for Principal Component Analysis

Abstract

The distribution of the latent vectors of a sample covariance matrix was found by T. W. Anderson [1] (1951) when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I.$ The asymptotic distribution for arbitrary $\Sigma$, also, was obtained by T. W. Anderson [3] in 1963. The elements of each latent vector are the coefficients of a principal component (with sum of squares of coefficients being unity). The object of the paper is to obtain the exact distribution of the latent vectors when the observations are obtained from bivariate normal distribution.