In mathematics, in particular functional analysis, the singular values, or snumbers of a compact operator T : X → Y acting between Hilbert spaces X and Y, are the square roots of nonnegative eigenvalues of the selfadjoint operator T^{*}T (where T^{*} denotes the adjoint of T).
The singular values are nonnegative 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 Minmax theorem).
In the case that 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 semiaxes 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 A = UΛU^{*}. Therefore, .
Most norms on Hilbert space operators studied are defined using snumbers. For example, the Ky Fanknorm is the sum of first k singular values, the trace norm is the sum of all singular values, and the Schatten norm is the pth root of the sum of the pth powers of the singular values. Note that each norm is defined only on a special class of operators, hence snumbers are useful in classifying different operators.
In the finitedimensional case, a matrix can always be decomposed in the form UΣV^{*}, 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.
YouTube Encyclopedic

1/5Views:157 788115 149358 297570580

✪ Singular Value Decomposition (the SVD)

✪ Lecture 47 — Singular Value Decomposition  Stanford University

✪ Singular matrices  Matrices  Precalculus  Khan Academy

✪ Linear Algebra  Lecture 42  The Singular Value Decomposition

✪ L8B: The Structured Singular Value
Transcription
Contents
Basic properties
For and .
Minmax theorem for singular values. Here is a subspace of of dimension .
Matrix transpose and conjugate do not alter singular values.
For any unitary
Relation to eigenvalues:
Inequalities about singular values
See also ^{[1]}.
Singular values of submatrices
For
 Let denote with one of its rows or columns deleted. Then
 Let denote with one of its rows and columns deleted. Then
 Let denote an submatrix of . Then
Singular values of
For
Singular values of
For
For ^{[2]}
Singular values and eigenvalues
For .
 See^{[3]}
 Assume . Then for :
 Weyl's theorem
 For .
 Weyl's theorem
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 nth snumber ^{[1]}:
This formulation made it possible to extend the notion of snumbers to operators in Banach space.
See also
 Condition number
 Cauchy interlacing theorem or Poincaré separation theorem
 Schur–Horn theorem
 Singular value decomposition
References
 ^ I. C. Gohberg and M. G. Krein. Introduction to the Theory of Linear Nonselfadjoint Operators. American Mathematical Society, Providence, R.I.,1969. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18.