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
The singular-value decomposition theorem says that M has a factorization of the form
- M = UΣ V*
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: Kn → Km 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*
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.
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