In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, in particular
functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defi ...
, the singular values, or ''s''-numbers of a
compact operator acting between
Hilbert space
In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...
s
and
, are the square roots of the (necessarily non-negative)
eigenvalue
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
s of the self-adjoint operator
(where
denotes the
adjoint of
).
The singular values are non-negative
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
s, usually listed in decreasing order (''σ''
1(''T''), ''σ''
2(''T''), …). The largest singular value ''σ''
1(''T'') is equal to the
operator norm of ''T'' (see
Min-max theorem
In linear algebra and functional analysis, the min-max theorem, or variational theorem, or Courant–Fischer–Weyl min-max principle, is a result that gives a variational characterization of eigenvalues of compact Hermitian operators o ...
).
If ''T'' acts on Euclidean space
, there is a simple geometric interpretation for the singular values: Consider the image by
of the
unit sphere
In mathematics, a unit sphere is simply a sphere of radius one around a given center. More generally, it is the set of points of distance 1 from a fixed central point, where different norms can be used as general notions of "distance". A unit ...
; this is an
ellipsoid
An ellipsoid is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.
An ellipsoid is a quadric surface; that is, a surface that may be defined as th ...
, and the lengths of its semi-axes are the singular values of
(the figure provides an example in
).
The singular values are the absolute values of the
eigenvalues
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
of a
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose :
The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...
''A'', because the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This is extremely useful be ...
can be applied to obtain unitary diagonalization of
as
. Therefore,
Most
norms on Hilbert space operators studied are defined using ''s''-numbers. For example, the
Ky Fan
Ky Fan (樊𰋀, , September 19, 1914 – March 22, 2010) was a Chinese-born American mathematician. He was a professor of mathematics at the University of California, Santa Barbara.
Biography
Fan was born in Hangzhou, the capital of Zhejiang ...
-''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
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
can always be decomposed in the form
, where
and
are
unitary matrices and
is a
rectangular diagonal matrix with the singular values lying on the diagonal. This is the
singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
.
Basic properties
For
, and
.
Min-max 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:
:
Relation to
trace:
:
.
If
is full rank, the product of singular values is
.
If
is full rank, the product of singular values is
.
If
is full rank, the product of singular values is
.
Inequalities about singular values
See also.
Singular values of sub-matrices
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 ''A'' + ''B''
For
#
#
Singular values of ''AB''
For
#
#
For
Singular values and eigenvalues
For
.
# See
# Assume
. Then for
:
##
Weyl's theorem In mathematics, Weyl's theorem or Weyl's lemma might refer to one of a number of results of Hermann Weyl. These include
* the Peter–Weyl theorem
* Weyl's theorem on complete reducibility, results originally derived from the unitarian trick on ...
## For
.
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:
[ I. C. Gohberg and M. G. Krein. Introduction to the Theory of Linear Non-selfadjoint Operators. American Mathematical Society, Providence, R.I.,1969. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18.]
:
This formulation made it possible to extend the notion of ''s''-numbers to operators in
Banach space
In mathematics, more specifically in functional analysis, a Banach space (pronounced ) is a complete normed vector space. Thus, a Banach space is a vector space with a metric that allows the computation of vector length and distance between vector ...
.
See also
*
Condition number
In numerical analysis, the condition number of a function measures how much the output value of the function can change for a small change in the input argument. This is used to measure how sensitive a function is to changes or errors in the inpu ...
*
Cauchy interlacing theorem or
Poincaré separation theorem
*
Schur–Horn theorem
In mathematics, particularly linear algebra, the Schur–Horn theorem, named after Issai Schur and Alfred Horn, characterizes the diagonal of a Hermitian matrix with given eigenvalues. It has inspired investigations and substantial generalization ...
*
Singular value decomposition
In linear algebra, the singular value decomposition (SVD) is a factorization of a real or complex matrix. It generalizes the eigendecomposition of a square normal matrix with an orthonormal eigenbasis to any \ m \times n\ matrix. It is re ...
References
{{Reflist
Operator theory
Singular value decomposition