Encyclopedia > Singular-value decomposition

  Article Content

Singular value decomposition

Redirected from Singular-value decomposition

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 Km and v in Kn such that
Mv = λu    and    M*u = λv
where 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&Sigma V*
where 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 Σi,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 u1,...,um of U are eigenvectors of MM* and are left singular vectors of M. The columns v1,...,vn of 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: KnKm that takes a vector x to Mx has a particularly simple description with respect to these orthonormal bases: we have T(vi) = di ui, for i = 1,...,min(m,n), where di is the i-th diagonal entry of D, and T(vi) = 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*

where 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 Km and Kn.

Add applications of singular value decomposition

See also: matrix decomposition

All Wikipedia text is available under the terms of the GNU Free Documentation License

  Search Encyclopedia

Search over one million articles, find something about almost anything!
  Featured Article
American Modern Library

... of contents 1 Best 20th Century Novel 1.1 Board Selections 1.2 Reader Selections 2 Best 20th Century Non-fiction 2.1 Board Selections 2.2 ...