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  "body": "\n## Definition\n\nLet $A$ be a real or complex $m \\times N$ [[matrix|0VD9ZSWWJG]] and let $1 \\le s \\le N$ be an integer.\nA vector $x$ is *$s$-sparse* if it has at most $s$ nonzero entries; $\\norm{\\cdot}_2$ is the Euclidean norm.\nThe *restricted isometry constant* $\\delta_s = \\delta_s(A)$ is the smallest $\\delta \\ge 0$ such that\n$$(1 - \\delta) \\norm{x}_2^2 \\le \\norm{Ax}_2^2 \\le (1 + \\delta) \\norm{x}_2^2 \\quad \\text{for every $s$-sparse } x \\in \\mathbb{K}^N ,$$\nwhere $\\mathbb{K}$ is $\\R$ or $\\C$.\nThe matrix $A$ has the *restricted isometry property* of order $s$ with constant $\\delta \\in (0, 1)$ if $\\delta_s(A) \\le \\delta$.\n\n## Remarks\n\nEquivalently, $\\delta_s = \\max_S \\norm{A_S^* A_S - I}_{2 \\to 2}$, where $S$ runs over the sets of at most $s$ column indices, $A_S$ keeps the columns in $S$, and the norm is the spectral [[matrix norm|54T6Y6DSY4]].\nSo $\\delta_s$ bounds how far the eigenvalues of all the Gram matrices $A_S^* A_S$ are from $1$.\nExample: if the columns of $A$ are orthonormal, then $\\norm{Ax}_2 = \\norm{x}_2$ for all $x$, so $\\delta_s = 0$ for every $s$.\nNon-example: suppose the first two columns of $A$ are equal.\nThen $x = e_1 - e_2$ is $2$-sparse with $Ax = 0$, so the left inequality needs $\\delta \\ge 1$, and $A$ has no restricted isometry property of order $2$.\nThe constants increase with the order: $\\delta_1 \\le \\delta_2 \\le \\cdots \\le \\delta_N$.\nIf $\\delta_{2s} \u003c 1$, then distinct $s$-sparse vectors have distinct images, since their difference is $2s$-sparse.\n",
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