Why covariance matrices are PSD

Why is any covariance matrix Σ\Sigma necessarily positive semidefinite?

Show hints (2)+
  1. A quadratic form in Σ\Sigma is a portfolio variance.
  2. Variance can never be negative.

Answer

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Because a⊤Σa=Var(a⊤X)≥0a^\top \Sigma a = \mathrm{Var}(a^\top X) \ge 0 for every aa

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