The Non-Singularity of Generalized Sample Covariance Matrices

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 σ(Xᵢ - X̄)(Xᵢ - X̄)' is positive definite w.p. 1 iff P X₁ ∈ F = 0 for every proper flat F ⊂ Rᵖ.