Singular value

In mathematics, in particular functional analysis, the singular values, or "s"-numbers of a compact operator acting between Hilbert spaces "X" and "Y", are the square roots of non-negative eigenvalues of the self-adjoint operator (where "T" denotes the adjoint of "T").
The singular values are non-negative real numbers, usually listed in decreasing order ("s"1("T"), "s"2("T"), …). The largest singular value "s"1("T") is equal to the operator norm of "T" (see Min-max theorem).
If "T" acts on euclidean space R"n", there is a simple geometric interpretation for the singular values: Consider the image by "T" of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of "T" (the figure provides an example in R"2").
The singular values are the absolute values of the eigenvalues of a normal matrix "A", because the spectral theorem can be applied to obtain unitary diagonalization of "A" as . Therefore, 
Most norms on Hilbert space operators studied are defined using "s"-numbers. For example, the Ky Fan-"k"-norm is the sum of first "k" singular values, the trace norm is the sum of all singular values, and the Schatten norm is the "p"th root of the sum of the "p"th powers 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 the finite-dimensional case, a matrix can always be decomposed in the form "U"Σ"V", where "U" and "V" are unitary matrices and Σ is a diagonal matrix with the singular values lying on the diagonal. This is the singular value decomposition.
Basic properties.
For formula_1, and formula_2.
Min-max theorem for singular values. Here formula_3 is a subspace of formula_4 of dimension formula_5.
Matrix transpose and conjugate do not alter singular values.
For any unitary formula_8
Relation to eigenvalues:
Inequalities about singular values.
See also.
Singular values of sub-matrices.
For formula_11
Singular values of "A" + "B".
For formula_22
Singular values of "AB".
For formula_25
For formula_22
Singular values and eigenvalues.
For formula_30.
History.
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, Allahverdiev proved the following characterization of the "n"th "s"-number : 
This formulation made it possible to extend the notion of "s"-numbers to operators in Banach space.