Inmathematics, in particularfunctional analysis, thesingular values of acompact operator${\displaystyle T:X\to Y}$ acting betweenHilbert spaces${\displaystyle X}$ and${\displaystyle Y}$, are the square roots of the (necessarily non-negative)eigenvalues of the self-adjoint operator${\displaystyle T^{\ast }T}$ (where${\displaystyle T^{\ast }}$ denotes theadjoint of${\displaystyle T}$).

The singular values are non-negativereal numbers, usually listed in decreasing order (σ₁(T),σ₂(T), …). The largest singular valueσ₁(T) is equal to theoperator norm ofT (seeMin-max theorem).

IfT acts on Euclidean space${\displaystyle {\mathbb{R}}^n}$, there is a simple geometric interpretation for the singular values: Consider the image by${\displaystyle T}$ of theunit sphere; this is anellipsoid, and the lengths of its semi-axes are the singular values of${\displaystyle T}$ (the figure provides an example in${\displaystyle {\mathbb{R}}^2}$).

The singular values are the absolute values of theeigenvalues of anormal matrixA, because thespectral theorem can be applied to obtain unitary diagonalization of${\displaystyle A}$ as${\displaystyle A=U\mathrm{\Lambda }{U}^{\ast }}$. Therefore,$\sqrt{A^{\ast }A}=\sqrt{U{\mathrm{\Lambda }}^{\ast }\mathrm{\Lambda }{U}^{\ast }}=U\left|\mathrm{\Lambda }\right|U^{\ast }$.

Mostnorms on Hilbert space operators studied are defined using singular values. For example, theKy Fan-k-norm is the sum of firstk singular values, the trace norm is the sum of all singular values, and theSchatten norm is thepth root of the sum of thepth 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, amatrix can always be decomposed in the form${\displaystyle \mathbf{U}\mathbf{\Sigma }{\mathbf{V}}^{\mathbf{\ast }}}$, where${\displaystyle \mathbf{U}}$ and${\displaystyle {\mathbf{V}}^{\mathbf{\ast }}}$ areunitary matrices and${\displaystyle \mathbf{\Sigma }}$ is arectangular diagonal matrix with the singular values lying on the diagonal. This is thesingular value decomposition.

For${\displaystyle A\in {\mathbb{C}}^{m\times n}}$, and${\displaystyle i=1,2,\dots ,min\{m,n\}}$.

Min-max theorem for singular values. Here${\displaystyle U:\dim (U)=i}$ is a subspace of${\displaystyle {\mathbb{C}}^n}$ of dimension${\displaystyle i}$.

Matrix transpose and conjugate do not alter singular values.

${\displaystyle {\sigma }_i(A)={\sigma }_i\left(A^{\mathsf{\text{T}}}\right)={\sigma }_i\left(A^{\ast }\right).}$

For any unitary${\displaystyle U\in {\mathbb{C}}^{m\times m},V\in {\mathbb{C}}^{n\times n}.}$

${\displaystyle {\sigma }_i(A)={\sigma }_i(UAV).}$

Relation to eigenvalues:

${\displaystyle {\sigma }_i^2(A)={\lambda }_i\left(A{A}^{\ast }\right)={\lambda }_i\left(A^{\ast }A\right).}$

Relation totrace:

${\displaystyle \sum _{i=1}^n{\sigma }_i^2=\text{tr}{A}^{\ast }A}$.

If${\displaystyle A^{\ast }A}$ has full rank, the product of singular values is${\displaystyle det\sqrt{A^{\ast }A}}$.

If${\displaystyle A{A}^{\ast }}$ has full rank, the product of singular values is${\displaystyle det\sqrt{A{A}^{\ast }}}$.

If${\displaystyle A}$ is square and has full rank, the product of singular values is${\displaystyle |detA|}$.

If${\displaystyle A}$ isnormal, then${\displaystyle \sigma (A)=|\lambda (A)|}$, that is, its singular values are the absolute values of its eigenvalues.

For a generic rectangular matrix${\displaystyle A}$, let be its augmented matrix. It has eigenvalues$\pm \sigma (A)$ (where$\sigma (A)$ are the singular values of$A$) and the remaining eigenvalues are zero. Let$A=U\mathrm{\Sigma }{V}^{\ast }$ be the singular value decomposition, then the eigenvectors of$\tilde{A}$ are$\left[\begin{array}{c}{\mathbf{u}}_i\\ \pm {\mathbf{v}}_i\end{array}\right]$ for${\displaystyle \pm {\sigma }_i}$^[1]: 52

The smallest singular value of a matrixA isσ_n(A). It has the following properties for a non-singular matrix A:

• The 2-norm of the inverse matrix A⁻¹ equals the inverseσ_n⁻¹(A).^{[2]: Thm.3.3}
• The absolute values of all elements in the inverse matrix A⁻¹ are at most the inverseσ_n⁻¹(A).^{[2]: Thm.3.3}

Intuitively, ifσ_n(A) is small, then the rows of A are "almost" linearly dependent. If it isσ_n(A) = 0, then the rows of A are linearly dependent and A is not invertible.

See also.^[3]

For${\displaystyle A\in {\mathbb{C}}^{m\times n}.}$

1. Let${\displaystyle B}$ denote${\displaystyle A}$ with one of its rowsor columns deleted. Then$${\displaystyle {\sigma }_{i+1}(A)\le {\sigma }_i(B)\le {\sigma }_i(A)}$$
2. Let${\displaystyle B}$ denote${\displaystyle A}$ with two of its rowsand columns deleted. Then$${\displaystyle {\sigma }_{i+2}(A)\le {\sigma }_i(B)\le {\sigma }_i(A)}$$
3. Let${\displaystyle B}$ denote an${\displaystyle (m-k)\times (n-\ell )}$ submatrix of${\displaystyle A}$. Then$${\displaystyle {\sigma }_{i+k+\ell }(A)\le {\sigma }_i(B)\le {\sigma }_i(A)}$$

For${\displaystyle A,B\in {\mathbb{C}}^{m\times n}}$

1. $${\displaystyle \sum _{i=1}^k{\sigma }_i(A+B)\le \sum _{i=1}^k({\sigma }_i(A)+{\sigma }_i(B)),k=min\{m,n\}}$$
2. $${\displaystyle {\sigma }_{i+j-1}(A+B)\le {\sigma }_i(A)+{\sigma }_j(B).i,j\in \mathbb{N},i+j-1\le min\{m,n\}}$$

For${\displaystyle A,B\in {\mathbb{C}}^{n\times n}}$

1. 
2. $${\displaystyle {\sigma }_n(A){\sigma }_i(B)\le {\sigma }_i(AB)\le {\sigma }_1(A){\sigma }_i(B)i=1,2,\dots ,n.}$$

For${\displaystyle A,B\in {\mathbb{C}}^{m\times n}}$^[4]$${\displaystyle 2{\sigma }_i(A{B}^{\ast })\le {\sigma }_i\left(A^{\ast }A+B^{\ast }B\right),i=1,2,\dots ,n.}$$

For${\displaystyle A\in {\mathbb{C}}^{n\times n}}$.

1. See^[5]$${\displaystyle {\lambda }_i\left(A+A^{\ast }\right)\le 2{\sigma }_i(A),i=1,2,\dots ,n.}$$
2. Assume${\displaystyle \left|{\lambda }_1(A)\right|\ge \cdots \ge \left|{\lambda }_n(A)\right|}$. Then for${\displaystyle k=1,2,\dots ,n}$:
   1. Weyl's theorem$${\displaystyle \prod _{i=1}^k\left|{\lambda }_i(A)\right|\le \prod _{i=1}^k{\sigma }_i(A).}$$
   2. For${\displaystyle p>0}$.$${\displaystyle \sum _{i=1}^k\left|{\lambda }_i^p(A)\right|\le \sum _{i=1}^k{\sigma }_i^p(A).}$$

This concept was introduced byErhard 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 thenth singular number:^[6]

This formulation made it possible to extend the notion of singular values to operators inBanach space. Note that there is a more general concept ofs-numbers, which also includes Gelfand and Kolmogorov width.

• Condition number
• Cauchy interlacing theorem orPoincaré separation theorem
• Schur–Horn theorem
• Singular value decomposition

• Operator theory
• Singular value decomposition

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