Sylvester's Criterion for Positive Definiteness (criterion)

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Positive definiteness is the condition that decides whether a quadratic form curves upwards, and Sylvester’s criterion turns it into a finite computation: look at the determinants of the leading blocks of the matrix, from the top left outwards.

The criterion is used wherever a second-order quantity has to be checked — a Hessian for a local minimum, a covariance for a Gaussian density, a preconditioner for a second-order optimiser — and each of those settings inherits the same test.

Criterion (Sylvester). A symmetric matrix \(M \in \mathbb{R}^{n \times n}\) is positive definite if and only if every leading principal minor is positive:

$$ \det M_k \gt 0, \qquad M_k = \left(M_{ij}\right)_{1 \leq i, j \leq k}, \qquad k = 1, \dots, n . $$

Two checks worth keeping alongside it: a positive definite matrix has all eigenvalues positive and hence a non-zero determinant, so the criterion is consistent with the spectral definition; and replacing \(\gt\) by \(\geq\) gives the test for positive semi-definiteness, where the minors need not all be positive, which is why the two cases are stated separately.

For a machine-learning problem the criterion is mostly used to diagnose failure: a Hessian whose minors change sign marks a saddle rather than a minimum, which is what makes the distinction visible in a computed matrix even when the loss curve is flat.