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  "body": "\n## Statement\n\nLet $A$ be an $n \\times n$ complex Hermitian matrix.\nFor $1 \\le k \\le n$, let $\\Delta_k$ be the determinant of the upper left $k \\times k$ submatrix of $A$.\nThen $A$ is [[positive definite|0KX66CZMBB]] if and only if $\\Delta_k \u003e 0$ for all $k = 1, \\dots, n$.\n\n## Remarks\n\nThe numbers $\\Delta_k$ are the *leading principal [[minors|0BT9BYWVRY]]* of $A$.\nFor $\\begin{pmatrix} a \u0026 b \\\\ b \u0026 c \\end{pmatrix}$ real, the test is $a \u003e 0$ and $ac - b^2 \u003e 0$.\nFor positive semidefinite matrices, the analogous test needs all principal minors, not only the leading ones, to be nonnegative.\n\n## Explanations\n\n### L1 human\n\nSylvester's criterion tests positive definiteness with $n$ determinants.\nWe check the top left $1 \\times 1$, $2 \\times 2$, and so on, up to the full matrix.\nAll of them must be positive.\n",
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