Positive Semidefiniteness of a Covariance Matrix

Write $v^T\Sigma v$ as the variance of a linear combination of the random vector's components.

Recall that variance can never be negative.

1. 1. Express the quadratic form as a variance

   For a random vector $X$ with covariance matrix $\Sigma$ and any fixed vector $v$, $v^T\Sigma v=\text{Var}(v^{T}X)$.

2. 2. Use nonnegativity of variance

   Since variance is always nonnegative, $v^T\Sigma v\ge 0$ for every $v$, which is exactly the definition of positive semidefiniteness.

Answer: For every vector v, v^T Sigma v = Var(v^T X) >= 0.