Covariance Matrices

Encoding variance and pairwise relationships in one object.

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

\Sigma_{ij} = \mathrm{Cov}(X_i, X_j)

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 $X \sim (\mu, \Sigma)$ ,

\mathrm{Var}(w^\top X) = w^\top \Sigma w

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