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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^\
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
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