Wikipedia / Singular value
Wikipedia / Singular value
In mathematics, in particular in functional analysis, the singular values of a compact operator acting between Hilbert spaces and , are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator denotes the adjoint of ).
The singular values are non-negative real numbers, usually listed in decreasing order is equal to the operator norm (see Min-max theorem).
acts on a Euclidean space (for example, of the unit sphere; this is an ellipsoid (the figure provides an example in ).
If is a normal matrix, then its singular values are the absolute values of its eigenvalues. Indeed, in this case, the spectral theorem as . Then, by definition, .
Most norms on Hilbert space operators studied are defined using singular values. For example, the Ky Fan singular values, the trace norm is the sum of all singular values, and the Schatten norm is the -th root of the sum of the -th powers of the singular values. Note that each norm is defined only on a special class of operators, hence singular values can be useful in classifying different operators.
In the finite-dimensional case, a matrix can always be decomposed in the form are unitary matrices is a rectangular diagonal matrix with the singular values lying on its diagonal. This is the singular value decomposition.
Basic properties
and :
Min-max theorem for singular values: is a subspace of ;
Matrix transpose and conjugate do not alter singular values:
.
and ,
.
Relation to eigenvalues:
.
Relation to trace:
.
has full rank, the product of singular values is .
has full rank, the product of singular values is .
is square and has full rank, the product of singular values is .
is normal, then , that is, its singular values are the absolute values of its eigenvalues.
For a generic rectangular matrix be its augmented matrix. eigenvalues of are the singular values of remaining eigenvalues of be the singular value decomposition of for are .
The smallest singular value
The smallest singular value of a matrix is . For a non-singular matrix , it has the following properties:
.
For all indices , .
Intuitively, if is small, then the rows of are "almost" linearly dependent. If it is , then the rows of are linearly dependent and is not invertible.
Inequalities about singular values
See also:
Singular values of sub-matrices
For ,
with one of its rows or
with two of its rows and
Singular values of A + B
For ,
Singular values of AB
For ,
For ,
Singular values and eigenvalues
For ,
See:
,
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 -th singular number:
This formulation made it possible to extend the notion of singular values to operators in Banach spaces.
Note that there is a more general concept of s-numbers, which also includes Gelfand and Kolmogorov width.
See also
References