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  "body": "\n## Definition\n\nLet $H$ and $K$ be complex [[Hilbert spaces|HM2HHE3J51]], and let $T \\colon H \\to K$ be a [[compact operator|M2NEP1K5AE]] with [[adjoint|6HSJBJ8TTQ]] $T^*$.\nLet $\\abs{T} = (T^* T)^{1/2}$ be the [[square root|3JY5HNFRMF]] of the [[positive operator|77020CY5ZW]] $T^* T$; it is compact and self-adjoint.\nList the nonzero [[eigenvalues|0BQAA2BK3Q]] of $\\abs{T}$ in decreasing order, each repeated as many times as the dimension of its eigenspace.\nIf there are only finitely many, append zeros.\nThe resulting sequence $s_1(T) \\ge s_2(T) \\ge \\cdots \\ge 0$ is the sequence of *singular values* of $T$.\n\n## Remarks\n\nThe list is well defined by the [[spectral theorem for compact self-adjoint operators|A1G25R99PY]], and $s_n(T) \\to 0$.\nOne has $s_1(T) = \\norm{T}$, and more generally\n$$s_n(T) = \\min \\set{\\norm{T - F} : F \\text{ of rank less than } n},$$\nthe distance from $T$ to the [[finite-rank operators|Z8FABSYQTS]] of rank less than $n$.\nThere are orthonormal families $(u_n)$ in $H$ and $(v_n)$ in $K$ such that $T x = \\sum_n s_n(T) \\ip{x}{u_n} v_n$ for all $x \\in H$, the Schmidt decomposition of $T$.\nExample: if $T$ is diagonal in an orthonormal basis $(e_n)$ of $\\ell^2$, with $T e_n = d_n e_n$ and $d_n \\to 0$, then $\\abs{T} e_n = \\abs{d_n} e_n$, so the singular values are the numbers $\\abs{d_n}$ arranged in decreasing order.\nNon-example: the identity of an infinite-dimensional Hilbert space is not compact, so it has no singular values in this sense; the eigenvalue $1$ of $\\abs{I} = I$ has infinite multiplicity.\nFor matrices, this is the usual notion of [[singular value|09VMDSRXWJ]].\n",
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