Singular value

In mathematics, in particular in functional analysis, the singular values of a compact operator ${\displaystyle \,T\!:X\rightarrow Y}$ acting between Hilbert spaces ${\displaystyle X}$ and ${\displaystyle Y}$, are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator ${\displaystyle T^{*}T}$ (where ${\displaystyle T^{*}}$ denotes the adjoint of ⁠${\displaystyle T}$⁠).

The singular values are non-negative real numbers, usually listed in decreasing order ⁠${\displaystyle {\big (}\sigma _{1}(T)\geq \sigma _{2}(T)\geq \dots {\big )}}$⁠. The largest singular value ${\displaystyle \sigma _{1}(T)}$ is equal to the operator norm of ${\displaystyle T}$ (see Min-max theorem).

If ${\displaystyle T}$ acts on a Euclidean space ⁠${\displaystyle \mathbb {R} ^{n}}$⁠, there is a simple geometric interpretation for the singular values: Consider the image by ${\displaystyle T}$ of the unit sphere; this is an ellipsoid, 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 the eigenvalues of a normal matrix ⁠${\displaystyle A}$⁠, because the spectral theorem can be applied to obtain unitary diagonalization of ${\displaystyle A}$ as ⁠${\displaystyle A=U\varLambda \,U^{*}}$⁠. Therefore, ${\textstyle {\sqrt {A^{*}A}}={\sqrt {U\varLambda ^{*}\varLambda \,U^{*}}}=U\left|\varLambda \right|U^{*}}$.

Most norms on Hilbert space operators studied are defined using singular values. For example, the Ky Fan ⁠${\displaystyle k}$⁠-norm is the sum of first ${\displaystyle k}$ singular values, the trace norm is the sum of all singular values, and the Schatten norm is the ⁠${\displaystyle p}$⁠-th root of the sum of the ⁠${\displaystyle p}$⁠-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 ⁠${\displaystyle \mathbf {U\Sigma V^{*}} }$⁠, where ${\displaystyle \mathbf {U} }$ and ${\displaystyle \mathbf {V^{*}} }$ are unitary matrices and ${\displaystyle \mathbf {\Sigma } }$ is a rectangular diagonal matrix with the singular values lying on the diagonal. This is the singular value decomposition.