Let $H \in \mathbb{R}^{n \times n}$ be a real symmetric matrix. Let $\Delta_k$ denote the determinant of the $k$-th leading principal minor of $H$ (the upper-left $k \times k$ submatrix), for $k = 1, 2, \ldots, n$. Then: - $H$ is **positive definite** if and only if $\Delta_k > 0$ for all $k = 1, \ldots, n$. - $H$ is **negative definite** if and only if $(-1)^k \Delta_k > 0$ for all $k = 1, \ldots, n$ (i.e. the signs alternate as $-, +, -, +, \ldots$).