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  "body": "\n## Definition\n\nLet $A$ be an $m \\times n$ complex matrix.\nThe matrix $A^* A$ is positive semidefinite, so its [[eigenvalues|0BQAA2BK3Q]] $\\lambda_1 \\ge \\cdots \\ge \\lambda_n \\ge 0$ are real and nonnegative.\nThe *singular values* of $A$ are the numbers\n$$\\sigma_i = \\sqrt{\\lambda_i} \\qquad (1 \\le i \\le n).$$\n\n## Remarks\n\nThe number of nonzero singular values is the rank of $A$.\nThe largest singular value is the operator norm $\\max_{\\norm{x} = 1} \\norm{Ax}$.\nFor a normal matrix, the singular values are the absolute values of the eigenvalues.\nSome authors list only the first $\\min(m, n)$ values.\n\n## Explanations\n\n### L1 human\n\nThe singular values of a matrix measure how much it stretches space in its main directions.\nThe unit sphere goes to an ellipsoid, and the singular values are the lengths of its semi-axes.\nUnlike eigenvalues, they exist for every matrix, also rectangular ones.\n",
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