Suppose

`M`is an

*m*-by-

*n*matrix whose entries come from the field

*K*, which is either the field of real numbers or the field of complex numbers. A non-negative real number λ is a

**singular value**for

*M*if there exist non-zero vectors

*u*in

*K*

^{m}and

*v*in

*K*

^{n}such that

*Mv*= λ*u*and*M*^{*}*u*= λ*v*

*M*

^{*}denotes the conjugate transpose of

*M*. The vectors

*u*and

*v*are called

**left-singular**and

**right-singular vectors**for λ, respectively.

The **singular-value decomposition** theorem says that `M` has a factorization of the form

*M*=*U*Σ*V*^{*}

`U`is an

*m*-by-

*m*unitary matrix over

*K*,

`V`is an

*n*-by-

*n*unitary matrix over

*K*, and Σ is an

*m*-by-

*n*diagonal matrix whose diagonal entries Σ

_{i,i}are non-negative real numbers. Such a factorization is called a

**singular-value decomposition**of

`M`.

In any such singular value decomposition, the diagonal entries of Σ are necessarily equal to the singular values of `M`.

The columns `u`_{1},...,`u _{m}` of

`U`are eigenvectors of

`MM`

^{*}and are left singular vectors of

*M*. The columns

`v`

_{1},...,

`v`of

_{n}`V`are eigenvectors of

`M`

^{*}

`M`and are right singular vectors of

`M`. Note however that different singular value decompositions of

*M*can contain different singular vectors.

The linear transformation *T*: *K*^{n} → *K*^{m} that takes a vector `x` to `Mx` has a particularly simple description with respect to these orthonormal bases: we have *T*(`v _{i}`) =

`d`, for

_{i}u_{i}*i*= 1,...,min(

*m*,

*n*), where

`d`is the

_{i}*i*-th diagonal entry of

*D*, and

*T*(

*v*

_{i}) = 0 for

*i*> min(

*m*,

*n*).

The number of non-zero singular values is equal to the rank *r* of *M*. These non-zero singular values are equal to the square roots of the non-zero eigenvalues of the positive semi-definite matrix `MM`^{*}, and also equal to the square roots of the non-zero eigenvalues of `M`^{*}`M`.

If we focus only on these *r* nonzero singular values, we can construct a singular-value decomposition of the following type:

*M*=*GDH*^{*}

*G*is an

*m*-by-

*r*orthonormal matrix over

*K*,

*H*is an

*n*-by-

*r*orthonormal matrix over

*K*and

*D*is an

*r*-by-

*r*diagonal matrix whose diagonal entries are positive real numbers.

The sum of the *k* largest singular values of *M* is a matrix norm, the **Ky Fan k-norm** of

`M`. The Ky Fan 1-norm is just the operator norm of

`M`as a linear operator with respect to the Euclidean norms of

*K*

^{m}and

*K*

^{n}.

*Add applications of singular value decomposition*