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A $k\times k$ matrix of covariances between all pairs of $k$ random variables. It is also called variance-covariance matrix or simply the covariance matrix.
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How does the matrix being centered change the eigenvalues/eigenvectors?
In case you are still interested in some answer to your question title, a paper by Paul Henoeine (2014) (link: http://arxiv.org/abs/1407.2904v1) may be of some relevance. In particular, Lemma 1 and T …