Editing Singular value decomposition (section)

{{Short description|Matrix decomposition}}
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[[File:Singular-Value-Decomposition.svg|thumb|Illustration of the singular value decomposition {{math|'''UΣV'''<sup>⁎</sup>}} of a real {{math|2&thinsp;×&thinsp;2}} matrix {{math|'''M'''}}.{{ubli
| '''Top:''' The action of {{math|'''M'''}}, indicated by its effect on the unit disc {{mvar|D}} and the two canonical unit vectors {{math|''e''<sub>1</sub>}} and {{math|''e''<sub>2</sub>}}.
| '''Left:''' The action of {{math|'''V'''<sup>⁎</sup>}}, a rotation, on {{math|''D''}}, {{math|''e''<sub>1</sub>}}, and {{math|''e''<sub>2</sub>}}.
| '''Bottom:''' The action of {{math|'''Σ'''}}, a scaling by the singular values {{math|''σ''<sub>1</sub>}} horizontally and {{math|''σ''<sub>2</sub>}} vertically.
| '''Right:''' The action of {{math|'''U'''}}, another rotation. 
}}
]]

In [[linear algebra]], the '''singular value decomposition''' ('''SVD''') is a [[Matrix decomposition|factorization]] of a [[real number|real]] or [[complex number|complex]] [[matrix (mathematics)|matrix]] into a rotation, followed by a rescaling followed by another rotation. It generalizes the [[eigendecomposition]] of a square [[normal matrix]] with an orthonormal eigenbasis to any {{tmath|m \times n}} matrix. It is related to the [[polar decomposition#Matrix polar decomposition|polar decomposition]].

Specifically, the singular value decomposition of an <math>m \times n</math> complex matrix {{tmath|\mathbf M}} is a factorization of the form <math>\mathbf{M} = \mathbf{U\Sigma V^*},</math> where {{tmath|\mathbf U}} is an {{tmath|m \times m}} complex [[unitary matrix]], <math>\mathbf \Sigma</math> is an <math>m \times n</math> [[rectangular diagonal matrix]] with non-negative real numbers on the diagonal, {{tmath|\mathbf V}} is an <math>n \times n</math> complex unitary matrix, and <math>\mathbf V^*</math> is the [[conjugate transpose]] of {{tmath|\mathbf V}}. Such decomposition always exists for any complex matrix.  If {{tmath|\mathbf M}} is real, then {{tmath|\mathbf U}} and {{tmath|\mathbf V}} can be guaranteed to be real [[orthogonal matrix|orthogonal]] matrices; in such contexts, the SVD is often denoted <math>\mathbf U \mathbf \Sigma \mathbf V^\mathrm{T}.</math>

The diagonal entries <math>\sigma_i = \Sigma_{i i}</math> of <math>\mathbf \Sigma</math> are uniquely determined by {{tmath|\mathbf M}} and are known as the [[singular value]]s of {{tmath|\mathbf M}}. The number of non-zero singular values is equal to the [[rank of a matrix|rank]] of {{tmath|\mathbf M}}. The columns of {{tmath|\mathbf U}} and the columns of {{tmath|\mathbf V}} are called left-singular vectors and right-singular vectors of {{tmath|\mathbf M}}, respectively. They form two sets of [[orthonormal basis|orthonormal bases]] {{tmath|\mathbf u_1, \ldots, \mathbf u_m}} and {{tmath|\mathbf v_1, \ldots, \mathbf v_n,}} and if they are sorted so that the singular values <math>\sigma_i</math> with value zero are all in the highest-numbered columns (or rows), the singular value decomposition can be written as

<math display=block>
\mathbf{M} = \sum_{i=1}^{r}\sigma_i\mathbf{u}_i\mathbf{v}_i^{*},
</math>

where <math>r \leq \min\{m,n\}</math> is the rank of {{tmath|\mathbf M.}}

The SVD is not unique. However, it is always possible to choose the decomposition such that the singular values <math>\Sigma_{i i}</math> are in descending order. In this case, <math>\mathbf \Sigma</math> (but not {{tmath|\mathbf U}} and {{tmath|\mathbf V}}) is uniquely determined by {{tmath|\mathbf M.}}

The term sometimes refers to the '''compact SVD''', a similar decomposition {{tmath|\mathbf M {{=}} \mathbf{U\Sigma V}^*}} in which {{tmath|\mathbf \Sigma}} is square diagonal of size {{tmath|r \times r,}} where {{tmath|r \leq \min\{m,n\} }} is the rank of {{tmath|\mathbf M,}} and has only the non-zero singular values. In this variant, {{tmath|\mathbf U}} is an {{tmath|m \times r}} [[semi-orthogonal matrix|semi-unitary matrix]] and <math>\mathbf{V}</math> is an {{tmath|n \times r}} [[semi-orthogonal matrix|semi-unitary matrix]], such that <math>\mathbf U^* \mathbf U = \mathbf V^* \mathbf V = \mathbf I_r.</math>

Mathematical applications of the SVD include computing the [[Moore–Penrose pseudoinverse|pseudoinverse]], matrix approximation, and determining the rank, [[range of a matrix|range]], and [[kernel (matrix)|null space]] of a matrix.  The SVD is also extremely useful in many areas of science, [[engineering]], and [[statistics]], such as [[signal processing]], [[least squares]] fitting of data, and [[process control]].