Singular value

In mathematics, in particular 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 (σ₁(T), σ₂(T), …). The largest singular value σ₁(T) is equal to the operator norm of T (see Min-max theorem).

If T 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 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 A, because the spectral theorem can be applied to obtain unitary diagonalization of ${\displaystyle A}$ as ${\displaystyle A=U\Lambda U^{*}}$. Therefore, ${\textstyle {\sqrt {A^{*}A}}={\sqrt {U\Lambda ^{*}\Lambda U^{*}}}=U\left|\Lambda \right|U^{*}}$.

Most norms on Hilbert space operators studied are defined using singular values. 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 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.