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  "body": "\n## Definition\n\nLet $A$ be an $n \\times n$ complex [[Hermitian matrix|0P17NGQQMQ]].\nThe matrix $A$ is *positive semidefinite* when\n$$x^* A x \\ge 0 \\qquad \\text{for all } x \\in \\C^n.$$\nFor a real symmetric $A$, the condition is $x^T A x \\ge 0$ for all $x \\in \\R^n$.\n\n## Remarks\n\n$A$ is positive semidefinite if and only if all its eigenvalues are nonnegative.\nEach [[positive definite|0KX66CZMBB]] matrix is positive semidefinite.\nThe matrix $\\begin{pmatrix} 1 \u0026 1 \\\\ 1 \u0026 1 \\end{pmatrix}$ is positive semidefinite but not positive definite.\nThe notation $A \\succeq 0$ is common.\n\n## Explanations\n\n### L1 human\n\nA positive semidefinite matrix never gives a negative value of $x^T A x$.\nIt may give zero for some nonzero vectors.\nCovariance matrices and Gram matrices are always of this kind.\n",
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