  This paper investigates a statistical procedure for testing the equality of
two independent estimated covariance matrices when the number of potentially
dependent data vectors is large and proportional to the size of the vectors,
that is, the number of variables. Inspired by the spike models used in random
matrix theory, we concentrate on the largest eigenvalues of the matrices in
order to determine significance. To avoid false rejections we must guard
against residual spikes and need a sufficiently precise description of the
behaviour of the largest eigenvalues under the null hypothesis. In this paper,
we lay a foundation by treating alternatives based on perturbations of order
$1$, that is, a single large eigenvalue. Our statistic allows the user to test
the equality of two populations. Future work will extend the result to
perturbations of order $k$ and demonstrate conservativeness of the procedure
for more general matrices.
