Covariance Matrices

For a vector of random variables $X = (X_1, \dots, X_k)$, the covariance matrix $\Sigma$ is

The diagonal holds the variances $\mathrm{Var}(X_i)$; the off-diagonal entries encode pairwise relationships.

Three properties matter:

• $\Sigma$ is symmetric: $\Sigma_{ij} = \Sigma_{ji}$.
• $\Sigma$ is positive semi-definite: $a^\top \Sigma a \ge 0$ for any vector $a$. This says portfolio variance can't be negative.
• For a linear combination $Y = a^\top X$, $\mathrm{Var}(Y) = a^\top \Sigma a$.

Portfolio variance is the canonical application: with weights $w$ and asset returns $X \sim (\mu, \Sigma)$,

Negative off-diagonal entries help — that's diversification. Positive off-diagonals hurt because correlated assets don't cancel each other's swings.