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| In [[mathematics]], in particular [[functional analysis]], the '''singular values''', or '''''s''-numbers''' of a [[compact operator]] {{nowrap|''T'' : ''X'' → ''Y''}} acting between [[Hilbert space]]s ''X'' and ''Y'', are the square roots of the [[eigenvalue]]s of the non-negative self-adjoint operator {{nowrap|''T*T'' : ''X'' → ''X''}} (where ''T*'' denotes the [[adjoint operator|adjoint]] of ''T'').
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| The singular values are non-negative [[real number]]s, usually listed in decreasing order (''s''<sub>1</sub>(''T''), ''s''<sub>2</sup>(''T''), …). If ''T'' is [[self-adjoint operator|self-adjoint]], then the largest singular value ''s''<sub>1</sub>(''T'') is equal to the [[operator norm]] of ''T'' (see [[Courant minimax principle]]).
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| [[File:Singular value decomposition.gif|thumb|right|280px|Visualisation of a [[singular value decomposition]] (SVD) of a 2-dimensional, real [[:en:Shear mapping|shearing matrix]] ''M''. First, we see the [[unit disc]] in blue together with the two [[standard basis|canonical unit vectors]]. We then see the action of ''M'', which distorts the disc to an [[ellipse]]. The SVD decomposes ''M'' into three simple transformations: a [[Rotation matrix|rotation]] ''V''<sup>*</sup>, a [[scaling (geometry)|scaling]] ''Σ'' along the rotated coordinate axes and a second rotation ''U''. ''Σ'' is a [[diagonal matrix]] containing in its diagonal the singular values of ''M'', which represent the lengths σ<sub>1</sub> and σ<sub>2</sub> of the [[Ellipse#Elements of an ellipse|semi-axes]] of the ellipse.]]
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| In the case that ''T'' acts on euclidean space '''R'''<sup>''n''</sup>, there is a simple geometric interpretation for the singular values: Consider the image by ''T'' of the [[N-sphere|unit sphere]]; this is an [[ellipsoid]], and its semi-axes are the singular values of ''T'' (the figure provides an example in '''R'''<sup>''2''</sup>).
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| In the case of a [[normal matrix]] ''A'', the [[spectral theorem]] can be applied to obtain unitary diagonalization of ''A'' as per {{nowrap|''A'' {{=}} ''U''Λ''U*''}}. Therefore, <math>\sqrt{A^*A}=U|\Lambda|U^*</math> and so the singular values are simply the absolute values of the [[eigenvalues]].
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| Most [[normed linear space|norms]] on Hilbert space operators studied are defined using ''s''-numbers. For example, the [[Ky Fan]]-''k''-norm is the sum of first ''k'' singular values, the trace norm is the sum of all singular values, and the [[Schatten norm]] is the ''p''th root of the sum of the ''p''th powers of the singular values. Note that each norm is defined only on a special class of operators, hence ''s''-numbers are useful in classifying different operators.
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| In the finite-dimensional case, a [[matrix (mathematics)|matrix]] can always be decomposed in the form ''UDW'', where ''U'' and ''W'' are [[unitary matrix|unitary matrices]] and ''D'' is a [[diagonal matrix]] with the singular values lying on the diagonal. This is the [[singular value decomposition]].
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| == History ==
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| This concept was introduced by [[Erhard Schmidt]] in 1907. Schmidt called singular values "eigenvalues" at that time. The name "singular value" was first quoted by Smithies in 1937. In 1957, Allahverdiev proved the following characterization of the ''n''th ''s''-number {{ref|1}}:
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| : <math>
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| s_n(T) = \inf\big\{\, \|T-L\| : L\text{ is an operator of finite rank }<n \,\big\}.
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| </math>
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| This formulation made it possible to extend the notion of ''s''-numbers to operators in [[Banach space]].
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| == See also ==
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| *[[Singular value decomposition]]
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| *[[Condition number]]
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| ==References== | |
| # {{note|1}}I. C. Gohberg and M. G. Krein. Introduction to the Theory of Linear Non-selfadjoint Operators. American Mathematical Society, Providence, R.I.,1969. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18.
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| [[Category:Operator theory]]
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| [[Category:Singular value decomposition]]
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I like my hobby Photography.
I also try to learn Korean in my spare time.
Feel free to visit my web site; traffic generation software