  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 propose some "invariant" theorems that allows us to extend
the test of arXiv:2002.12741 for perturbation of order $1$ to some general
tests for order $k$. The statistics introduced in this paper allow the user to
test the equality of two populations based on high-dimensional multivariate
data.
  Simulations show that these tests have more power of detection than standard
multivariate approaches.
