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  "body": "\n## Definition\n\nLet $A$ be an $n \\times n$ complex [[Hermitian matrix|0P17NGQQMQ]].\nThe matrix $A$ is *positive definite* when\n$$x^* A x \u003e 0 \\qquad \\text{for all } x \\in \\C^n \\text{ with } x \\ne 0.$$\nA real [[symmetric matrix|0VR671PHE5]] $A$ is positive definite when $x^T A x \u003e 0$ for all $x \\in \\R^n$ with $x \\ne 0$.\n\n## Remarks\n\nFor a Hermitian $A$, the number $x^* A x$ is always real.\nThe identity matrix and $\\begin{pmatrix} 2 \u0026 -1 \\\\ -1 \u0026 2 \\end{pmatrix}$ are positive definite.\n$A$ is positive definite if and only if all its eigenvalues are positive.\n$A$ is positive definite if and only if $\\langle x, y \\rangle = y^* A x$ is an inner product on $\\C^n$.\n\n## Explanations\n\n### L1 human\n\nA positive definite matrix gives a positive number $x^T A x$ for every nonzero vector $x$.\nSuch matrices behave like positive numbers in many ways: they have square roots and positive determinants.\nThey appear as Hessians at minima and as covariance matrices.\n",
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