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  "body": "\n## Statement\n\nLet $A$ be a [[positive definite|0KX66CZMBB]] $n \\times n$ complex Hermitian matrix.\nThen there is exactly one [[lower triangular matrix|00J3KZP8Y0]] $L$ with positive real diagonal entries such that\n$$A = L L^*.$$\nIf $A$ is real, then $L$ is real.\n\n## Remarks\n\nThe factor $L$ is the *Cholesky factor* of $A$.\nFor example, $\\begin{pmatrix} 4 \u0026 2 \\\\ 2 \u0026 5 \\end{pmatrix} = \\begin{pmatrix} 2 \u0026 0 \\\\ 1 \u0026 2 \\end{pmatrix} \\begin{pmatrix} 2 \u0026 1 \\\\ 0 \u0026 2 \\end{pmatrix}$.\nIt needs about half the work of an LU decomposition, and it solves systems $Ax = b$ with positive definite $A$.\nA positive semidefinite matrix also has a decomposition $A = L L^*$, but $L$ need not be unique.\n\n## Explanations\n\n### L1 human\n\nA positive definite matrix is the product of a lower triangular matrix and its conjugate transpose.\nThis is like writing a positive number as the square of its positive square root.\nIt is the fastest standard way to solve linear systems with such a matrix.\n",
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