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The classical methods of multivariate analysis are based on the eigenvaluesof one or two sample covariance matrices. In many applications of thesemethods, for example to high dimensional data, it is natural to consideralternative hypotheses which are a low rank departure from the null hypothesis.For rank one alternatives, this note provides a representation for the jointeigenvalue density in terms of a single contour integral. This will be of usefor deriving approximate distributions for likelihood ratios and linearstatistics used in testing.

Joint density of eigenvalues in spiked multivariate models