Singular value decomposition (original) (raw)

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_

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_Σ 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 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 u1,...,um of U are eigenvectors of MM* and are left singular vectors of M. The columns v1,...,vnof 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(vi) = di ui, for i = 1,...,min(m,n), where di is the _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*

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 K m and K n.

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