Linear algebra

Positive semi-definiteness of a covariance matrix

Tests whether a proposed correlation matrix can exist, in one line.

w^{\top}\Sigma w = \mathrm{Var}(w^{\top}X) \ge 0 \quad \Rightarrow \quad \Sigma \succeq 0

Where

$\rho \ge -\tfrac{1}{n-1}$: The floor for $n$ pairwise equicorrelated assets.
$\sum_i\lambda_i = \mathrm{tr}(\Sigma)$: Eigenvalues sum to the total variance.

Assumptions

A *sample* covariance matrix from $T$ observations has rank at most $T-1$ , so it is singular when $n \ge T$ .

Sanity check. A negative eigenvalue means a portfolio with negative variance, which cannot exist.

Where this is taught

PCA, covariance matrices and what an eigenvalue is telling you · LA · SVD and dimensionality reduction