S-number
Encyclopedia
In mathematics
, in particular functional analysis
, the singular values, or s-numbers of a compact operator
acting between Hilbert space
s X and Y, are the square roots of the eigenvalues of the nonnegative self-adjoint operator (where T* denotes the adjoint of T).
The singular values are nonnegative real number
s, usually listed in decreasing order (s1(T), s2(T), …). If T is self-adjoint
, then the largest singular value s1(T) is equal to the operator norm
of T (see Courant minimax principle
).
In the case that T acts on euclidean space Rn, there is a simple geometric interpretation for the singular values: Consider the image by T of the unit sphere; this is an ellipsoid, and its semi-axes are the singular values of T (the figure provides an example in R2).
In the case of a normal matrix
A, the spectral theorem
can be applied to obtain unitary diagonalization of A as per . Therefore,
and so the singular values are simply the absolute values of the eigenvalues.
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 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 s-numbers are useful in classifying different operators.
In the finite-dimensional case, a matrix
can always be decomposed in the form UDW, where U and W are unitary matrices and D is a diagonal matrix
with the singular values lying on the diagonal. This is the singular value decomposition
.
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 the following characterization of the nth s-number:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in particular functional analysis
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 and the linear operators acting upon these spaces and respecting these structures in a suitable sense...
, the singular values, or s-numbers of a compact operator
Compact operator
In functional analysis, a branch of mathematics, a compact operator is a linear operator L from a Banach space X to another Banach space Y, such that the image under L of any bounded subset of X is a relatively compact subset of Y...
acting between Hilbert space
Hilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s X and Y, are the square roots of the eigenvalues of the nonnegative self-adjoint operator (where T* denotes the adjoint of T).
The singular values are nonnegative real number
Real number
In mathematics, a real number is a value that represents a quantity along a continuum, such as -5 , 4/3 , 8.6 , √2 and π...
s, usually listed in decreasing order (s1(T), s2(T), …). If T is self-adjoint
Self-adjoint operator
In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is an operator that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose...
, then the largest singular value s1(T) is equal to the operator norm
Operator norm
In mathematics, the operator norm is a means to measure the "size" of certain linear operators. Formally, it is a norm defined on the space of bounded linear operators between two given normed vector spaces.- Introduction and definition :...
of T (see Courant minimax principle
Courant minimax principle
In mathematics, the Courant minimax principle gives the eigenvalues of a real symmetric matrix. It is named after Richard Courant.-Introduction:The Courant minimax principle gives a condition for finding the eigenvalues for a real symmetric matrix...
).
In the case that T acts on euclidean space Rn, there is a simple geometric interpretation for the singular values: Consider the image by T of the unit sphere; this is an ellipsoid, and its semi-axes are the singular values of T (the figure provides an example in R2).
In the case of a normal matrix
Normal matrix
A complex square matrix A is a normal matrix ifA^*A=AA^* \ where A* is the conjugate transpose of A. That is, a matrix is normal if it commutes with its conjugate transpose.If A is a real matrix, then A*=AT...
A, the spectral theorem
Spectral theorem
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized...
can be applied to obtain unitary diagonalization of A as per . Therefore,
and so the singular values are simply the absolute values of the eigenvalues.Most norms on Hilbert space operators studied are defined using s-numbers. For example, the Ky Fan
Ky Fan
Ky Fan was an American mathematician and Emeritus Professor of Mathematics at the University of California, Santa Barbara .-Biography:...
-k-norm is the sum of first k singular values, the trace norm is the sum of all singular values, and the Schatten norm
Schatten norm
In mathematics, specifically functional analysis, the Schatten norm arises as a generalization of p-integrability similar to the trace class norm and the Hilbert–Schmidt norm.-Definition:...
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 s-numbers are useful in classifying different operators.
In the finite-dimensional case, a matrix
Matrix (mathematics)
In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions. The individual items in a matrix are called its elements or entries. An example of a matrix with six elements isMatrices of the same size can be added or subtracted element by element...
can always be decomposed in the form UDW, where U and W are unitary matrices and D is a diagonal matrix
Diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero...
with the singular values lying on the diagonal. This is the singular value decomposition
Singular value decomposition
In linear algebra, the singular value decomposition is a factorization of a real or complex matrix, with many useful applications in signal processing and statistics....
.
History
This concept was introduced by Erhard SchmidtErhard Schmidt
Erhard Schmidt was a German mathematician whose work significantly influenced the direction of mathematics in the twentieth century. He was born in Tartu, Governorate of Livonia . His advisor was David Hilbert and he was awarded his doctorate from Georg-August University of Göttingen in 1905...
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 the following characterization of the nth s-number:
-
This formulation made it possible to extend the notion of s-numbers to operators in Banach spaceBanach spaceIn mathematics, Banach spaces is the name for complete normed vector spaces, one of the central objects of study in functional analysis. A complete normed vector space is a vector space V with a norm ||·|| such that every Cauchy sequence in V has a limit in V In mathematics, Banach spaces is the...
.