Positive Definite Matrices

A symmetric matrix $A$ is positive definite (PD) if $x^T A x > 0$ for every nonzero vector $x$. It is positive semi-definite (PSD) if $x^T A x \ge 0$. Equivalent conditions (any one implies the others for symmetric $A$): - All eigenvalues are positive (PD) or non-negative (PSD) - All leading principal minors are positive (Sylvester's criterion) - $A = B^T B$ for some full-rank matrix $B$ (Cholesky-like factorization) - All pivots in Gaussian elimination are positive Concrete example: Is $A = \begin{pmatrix} 4 & 2 \\ 2 & 3 \end{pmatrix}$ positive definite? Check: $\text{tr}(A) = 7 > 0$ and $\det(A) = 12 - 4 = 8 > 0$. Since both eigenvalues are positive (their sum and product are both positive), $A$ is PD. Why it matters in finance: Every valid covariance matrix is PSD. Portfolio variance is $w^T \Sigma w$, which must be non-negative for any portfolio weights $w$. If $\Sigma$ is PD, every non-degenerate portfolio has strictly positive risk. A correlation matrix with an eigenvalue $\le 0$ signals a data problem or an impossible correlation structure. When to use: Verifying that a given matrix can serve as a covariance matrix, checking convexity of quadratic optimization problems (the Hessian must be PSD for convexity), and understanding when $x^T A x$ achieves a unique minimum.