In mathematics, in particular functional analysis, singular values, or s-numbers of an bounded operatorT acting on a Hilbert space are defined as the eigenvalues of (T*T)1/2. They are nonnegative real numbers, usually listed in decreasing order s1(T), s2(T), ... . The largest singular value s1(T) is simply the operator norm of T.
This concept was introduced by Erhard Schmidt in 1907. Schmidt called singular values "eigenvalues" at that time. The name "singular value" was first quoted by Smithies in 1937. In 1957, Allakhverdiev proved that the nth s-number
This formulation made it possible to extend the notion of s-numbers to operators in Banach space.
Most norms on Hilberst space operators studied are defined using s-numbers. For example, Ky Fan-k-norm is the sum of first k singular values, trace norm is the sum of all singular values, the Schatten-p-norm is the pth root of the sum of the pth power of the singular values. Note that each norm is defined only on a special class of operators, hence s-numbers are useful in classifying different operators.
In finite-dimensional case, a matrix could always decomposed into the from UDW, where U and W are unitary matrices and D is a diagonal matrix with the singular values lying on the diagonal. It is called the singular value decomposition.
The singularvalues and the structure of the singular vectors depend on the length of the simulation interval, the meteorological data, the location of the optimization region and the selection of optimization species, the choice of error norms, and the size of the optimization region.
While in the study of atmospheric dynamics the dominant singular vectors are associated with unstable modes, in the study of chemical transport systems the dominant singular vectors are useful to describe the uncertainty in a limited subdomain (e.g., where the model prediction needs to be improved).
Singular vectors are defined as the directions of maximal error growth, that is, the vectors s^sub k^(t^sub 0^) that maximize the ratio σ^sup 2^ in (7).
In numerical linear algebra the singularvalues can be used to determine the effective rank of a matrix, as rounding error may lead to small but non-zero singularvalues in a rank deficient matrix.
The square root of the sum of squares of the singularvalues is the Frobenius norm of M.
The singularvalue decomposition is used for computing the pseudoinverse of a matrix.
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