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Let X = (X₁, , Xₙ) where the Xᵢ: p 1 are independent random vectors, and let A: n n be positive semi-definite symmetric. This paper establishes necessary and sufficient conditions that the random matrix XAX' be positive definite w. p. 1. The results are applied to cases where A has a particular form or X₁, , Xₙ are i. i. d. In particular, it is shown that in the i. i. d. case, the sample covariance matrix (Xᵢ - X) (Xᵢ - X) ' is positive definite w. p. 1 iff P X₁ F = 0 for every proper flat F Rᵖ.
Eaton et al. (Sun,) studied this question.