In _{1}(''T''), ''s''_{2}(''T''), …). The largest singular value ''s''_{1}(''T'') is equal to the ^{''n''}, there is a simple geometric interpretation for the singular values: Consider the image by ''T'' of the N-sphere, 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 normed linear space, 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 (mathematics), matrix can always be decomposed in the form ''U''Σ''V'', where ''U'' and ''V'' are unitary matrix, unitary matrices and Σ is a diagonal matrix with the singular values lying on the diagonal. This is the singular value decomposition.

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, in particular functional analysis
200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional analysis.
Functional analysis is a branch of mathemat ...

, the singular values, or ''s''-numbers of a compact operator
In functional analysis
Image:Drum vibration mode12.gif, 200px, One of the possible modes of vibration of an idealized circular drum head. These modes are eigenfunctions of a linear operator on a function space, a common construction in functional a ...

acting between Hilbert space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

s ''X'' and ''Y'', are the square roots of non-negative eigenvalue
In linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and their representations in vector spaces an ...

s of the self-adjoint operator (where ''T'' denotes the adjoint of ''T'').
The singular values are non-negative real number
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

s, usually listed in decreasing order (''s''operator norm
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

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 and eigenvectors, eigenvalues of ...

).
If ''T'' acts on euclidean space RBasic properties

For $A\; \backslash in\; \backslash mathbb^$, and $i\; =\; 1,2,\; \backslash ldots,\; \backslash min\; \backslash $. Min-max theorem#Min-max principle for singular values, Min-max theorem for singular values. Here $U:\; \backslash dim(U)\; =\; i$ is a subspace of $\backslash mathbb^n$ of dimension $i$. :$\backslash begin\; \backslash sigma\_i(A)\; \&=\; \backslash min\_\; \backslash max\_\; \backslash left\backslash ,\; Ax\; \backslash right\backslash ,\; \_2.\; \backslash \backslash \; \backslash sigma\_i(A)\; \&=\; \backslash max\_\; \backslash min\_\; \backslash left\backslash ,\; Ax\; \backslash right\backslash ,\; \_2.\; \backslash end$ Matrix transpose and conjugate do not alter singular values. :$\backslash sigma\_i(A)\; =\; \backslash sigma\_i\backslash left(A^\backslash textsf\backslash right)\; =\; \backslash sigma\_i\backslash left(A^*\backslash right)\; =\; \backslash sigma\_i\backslash left(\backslash bar\backslash right).$ For any unitary $U\; \backslash in\; \backslash mathbb^,\; V\; \backslash in\; \backslash mathbb^.$ :$\backslash sigma\_i(A)\; =\; \backslash sigma\_i(UAV).$ Relation to eigenvalues: :$\backslash sigma\_i^2(A)\; =\; \backslash lambda\_i\backslash left(AA^*\backslash right)\; =\; \backslash lambda\_i\backslash left(A^*A\backslash right).$Inequalities about singular values

See also.Singular values of sub-matrices

For $A\; \backslash in\; \backslash mathbb^.$ # Let $B$ denote $A$ with one of its rows ''or'' columns deleted. Then #: $\backslash sigma\_(A)\; \backslash leq\; \backslash sigma\_i\; (B)\; \backslash leq\; \backslash sigma\_i(A)$ # Let $B$ denote $A$ with one of its rows ''and'' columns deleted. Then #: $\backslash sigma\_(A)\; \backslash leq\; \backslash sigma\_i\; (B)\; \backslash leq\; \backslash sigma\_i(A)$ # Let $B$ denote an $(m-k)\backslash times(n-l)$ submatrix of $A$. Then #: $\backslash sigma\_(A)\; \backslash leq\; \backslash sigma\_i\; (B)\; \backslash leq\; \backslash sigma\_i(A)$Singular values of ''A'' + ''B''

For $A,\; B\; \backslash in\; \backslash mathbb^$ # $\backslash sum\_^\; \backslash sigma\_i(A\; +\; B)\; \backslash leq\; \backslash sum\_^\; \backslash sigma\_i(A)\; +\; \backslash sigma\_i(B),\; \backslash quad\; k=\backslash min\; \backslash $ # $\backslash sigma\_(A\; +\; B)\; \backslash leq\; \backslash sigma\_i(A)\; +\; \backslash sigma\_j(B).\; \backslash quad\; i,j\backslash in\backslash mathbb,\backslash \; i\; +\; j\; -\; 1\; \backslash leq\; \backslash min\; \backslash $Singular values of ''AB''

For $A,\; B\; \backslash in\; \backslash mathbb^$ # $\backslash begin\; \backslash prod\_^\; \backslash sigma\_i(A)\; \backslash sigma\_i(B)\; \&\backslash leq\; \backslash prod\_^\; \backslash sigma\_i(AB)\; \backslash \backslash \; \backslash prod\_^k\; \backslash sigma\_i(AB)\; \&\backslash leq\; \backslash prod\_^k\; \backslash sigma\_i(A)\; \backslash sigma\_i(B),\; \backslash \backslash \; \backslash sum\_^k\; \backslash sigma\_i^p(AB)\; \&\backslash leq\; \backslash sum\_^k\; \backslash sigma\_i^p(A)\; \backslash sigma\_i^p(B),\; \backslash end$ # $\backslash sigma\_n(A)\; \backslash sigma\_i(B)\; \backslash leq\; \backslash sigma\_i\; (AB)\; \backslash leq\; \backslash sigma\_1(A)\; \backslash sigma\_i(B)\; \backslash quad\; i\; =\; 1,\; 2,\; \backslash ldots,\; n.$ For $A,\; B\; \backslash in\; \backslash mathbb^$ :$2\; \backslash sigma\_i(A\; B^*)\; \backslash leq\; \backslash sigma\_i\; \backslash left(A^*\; A\; +\; B^*\; B\backslash right),\; \backslash quad\; i\; =\; 1,\; 2,\; \backslash ldots,\; n.$Singular values and eigenvalues

For $A\; \backslash in\; \backslash mathbb^$. # See #: $\backslash lambda\_i\backslash left(A\; +\; A^*\backslash right)\; \backslash leq\; 2\; \backslash sigma\_i(A),\; \backslash quad\; i\; =\; 1,\; 2,\; \backslash ldots,\; n.$ # Assume $\backslash left,\; \backslash lambda\_1(A)\backslash \; \backslash geq\; \backslash cdots\; \backslash geq\; \backslash left,\; \backslash lambda\_n(A)\backslash $. Then for $k\; =\; 1,\; 2,\; \backslash ldots,\; n$: ## Weyl's inequality#Weyl's inequality in matrix theory, Weyl's theorem ##: $\backslash prod\_^k\; \backslash left,\; \backslash lambda\_i(A)\backslash \; \backslash leq\; \backslash prod\_^\; \backslash sigma\_i(A).$ ## For $p>0$. ##: $\backslash sum\_^k\; \backslash left,\; \backslash lambda\_i^p(A)\backslash \; \backslash leq\; \backslash sum\_^\; \backslash sigma\_i^p(A).$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:Israel Gohberg, I. C. Gohberg and Mark Krein, 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. : $s\_n(T)\; =\; \backslash inf\backslash big\backslash .$ This formulation made it possible to extend the notion of ''s''-numbers to operators in Banach space.See also

*Condition number *Min-max theorem#Cauchy interlacing theorem, Cauchy interlacing theorem or Poincaré separation theorem *Schur–Horn theorem *Singular value decompositionReferences

{{Reflist Operator theory Singular value decomposition