Sylvester's criterion characterizes positive definite (PD) and positive semidefinite (PSD) matrices without the need of eigendecomposition. It states that a symmetric matrix is PD if and only if all of its leading principal minors are positive, and a symmetric matrix is PSD if and only if all of its principal minors are nonnegative. For an $m\times m$ symmetric matrix, Sylvester's criterion requires computing $m$ and $2^m-1$ determinants to verify it is PD and PSD, respectively. Therefore, it is less useful for PSD matrices due to the exponential growth in the number of principal submatrices as the matrix dimension increases. We provide a stronger Sylvester's criterion for PSD matrices which only requires to verify the nonnegativity of $m(m+1)/2$ determinants. Based on the new criterion, we provide a method to derive elementwise criteria for PD and PSD matrices. We illustrate the applications of our results in PD or PSD matrix completion and highlight their statistics applications via nonlinear semidefinite program.

翻译：西尔维斯特准则无需特征分解即可刻画正定（PD）矩阵与半正定（PSD）矩阵的性质。该准则指出：对称矩阵正定当且仅当其所有顺序主子式为正；对称矩阵半正定当且仅当其所有主子式非负。对于 $m\times m$ 对称矩阵，验证其正定性与半正定性分别需要计算 $m$ 个和 $2^m-1$ 个行列式。由于矩阵维度增加时主子矩阵数量呈指数增长，该准则对半正定矩阵的实用性较低。本文提出适用于半正定矩阵的强化西尔维斯特准则，仅需验证 $m(m+1)/2$ 个行列式的非负性。基于新准则，我们提出推导正定与半正定矩阵逐元素判据的方法。通过非线性半定规划，我们展示了该成果在正定/半正定矩阵补全问题中的应用，并阐明了其在统计学中的潜在价值。