Editing Singular value (section)

{{Short description|Square roots of the eigenvalues of the self-adjoint operator}}
In [[mathematics]], in particular [[functional analysis]], the '''singular values''' of a [[compact operator]] <math>T: X \rightarrow Y</math> acting between [[Hilbert space]]s  <math>X</math> and <math>Y</math>, are the square roots of the (necessarily non-negative) [[eigenvalue]]s of the self-adjoint operator <math>T^*T</math> (where <math>T^*</math> denotes the [[adjoint operator|adjoint]] of <math>T</math>).

The singular values are non-negative [[real number]]s, usually listed in decreasing order (''σ''<sub>1</sub>(''T''), ''σ''<sub>2</sub>(''T''), …).  The largest singular value ''σ''<sub>1</sub>(''T'') is equal to the [[operator norm]] of ''T'' (see [[Min-max theorem#Min-max principle for singular values|Min-max theorem]]).

[[File:Singular value decomposition.gif|thumb|right|280px|Visualization 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|*}}, a [[scaling (geometry)|scaling]] Σ along the rotated coordinate axes and a second rotation ''U''. Σ is a (square, in this example) [[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.]]

If ''T'' acts on Euclidean space <math>\Reals ^n</math>, there is a simple geometric interpretation for the singular values: Consider the image by <math>T</math> of the [[N-sphere|unit sphere]]; this is an [[ellipsoid]], and the lengths of its semi-axes are the singular values of <math>T</math> (the figure provides an example in <math>\Reals^2</math>).

The singular values are the absolute values of the [[eigenvalues]] of a [[normal matrix]] ''A'', because the [[spectral theorem]] can be applied to obtain unitary diagonalization of <math>A</math> as <math>A = U\Lambda U^*</math>. Therefore, {{nowrap|<math display="inline">\sqrt{A^* A} = \sqrt{U \Lambda^* \Lambda U^*} = U \left| \Lambda \right| U^*</math>.}}

Most [[normed linear space|norms]] on Hilbert space operators studied are defined using singular values.  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 singular values can be useful in classifying different operators.

In the finite-dimensional case, a [[matrix (mathematics)|matrix]] can always be decomposed in the form <math>\mathbf{U\Sigma V^*}</math>, where <math>\mathbf{U}</math> and <math>\mathbf{V^*}</math> are [[unitary matrix|unitary matrices]] and <math>\mathbf{\Sigma}</math> is a [[rectangular diagonal matrix]] with the singular values lying on the diagonal. This is the [[singular value decomposition]].