On the Distribution of the Largest Latent Root and the Corresponding Latent Vector for Principal Component Analysis

Abstract

The distribution of the latent vectors of a sample covariance matrix was found by T. W. Anderson [1] in 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 exact distribution of the latent vectors of a sample covariance matrix has been described by the author [10] in 1965 when the observations are obtained from a bi-variate normal distribution. The elements of each latent vector are the coefficients of a principal component (with sum of squares of coefficients being unity), and the corresponding latent root is the variance of the principal component. In this paper, the exact distribution of the latent vector corresponding to the largest latent root of a sample covariance matrix is given when the observations are from a multivariate normal distribution whose population covariance matrix is arbitrary $\Sigma$, and the distribution of the largest latent root is given when the population covariance matrix is a scalar matrix, $\Sigma = \sigma^2I$.