Interpreting the First Eigenvalue of a Correlation Matrix

Abstract

The first eigenvalue of a correlation matrix indicates the maximum amount of the variance of the variables which can be accounted for with a linear model by a single underlying factor. When all correlations are positive, this first eigenvalue is approximately a linear function of the average correlation among the variables. While that is not true when not all the correlations are positive, in the general case the first eigenvalue is approximately equal to a lower bound derived in the paper. That lower bound is based on the maximum average correlation over reversals of variables and over subsets of the variables. Regression tests show these linear approximations are very accurate. The first eigenvalue measures the primary cluster in the matrix, its number of variables and average correlation.