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What does it mean when you say "Covariance matrix are highly ellipsoidal"? I am reading Regularized Discriminant Analysis" by Jerome H. Friedman 1989 and it uses this term very often "Covariance matrices are all equal and Highly ellipsoidal "?

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  • $\begingroup$ It's quite useless just to say you're reading "a paper". Perhaps you're more likely to get an answer if you tell us which paper it is. $\endgroup$ Jun 19, 2014 at 21:04
  • $\begingroup$ Sorry for incomplete information. The paper is titled "Regularized Discriminant Analysis" by Jerome H. Friedman 1989, published in Journal of the American Statistical Association $\endgroup$
    – Spandyie
    Jun 20, 2014 at 16:04

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The covariance matrix of a set of variables is always symmetric by definition, which means that it can be diagonalized by an orthogonal matrix.

That being said, we can visualize the matrix as an ellipsoid whose axes have sizes given by it eigenvalues, and that has been rotated by the corresponding orthogonal matrix.

In this context the term ellipsoidal is being used in opposition to spherical. Being highly ellipsoidal means that the eigenvalues have very different magnitudes.

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