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We study the facial structure of the set En × n of correlation matrices (i.e., the positive semidefinite matrices with diagonal entries equal to 1). In particular, we determine the possible dimensions for a face, as well as for a polyhedral face, of En × n. It turns out that the spectrum of face dimensions is lacunary and that En × n has polyhedral faces of dimension up to ≈ √2n. As an application, we describe in detail the faces of E4 × 4. We also discuss results related to optimization over En × n.
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Laurent et al. (1996) studied this question.
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