In mathematics, in particular in functional analysis, the singular values of a compact operator T : X → Y {\displaystyle \,T\!:X\rightarrow Y} acting between Hilbert spaces X {\displaystyle X} and Y {\displaystyle Y} , are the square roots of the (necessarily non-negative) eigenvalues of the self-adjoint operator T ∗ T {\displaystyle T^{*}T} (where T ∗ {\displaystyle T^{*}} denotes the adjoint of T {\displaystyle T} ). The singular values are non-negative real numbers, usually listed in decreasing order ( σ 1 ( T ) ≥ σ 2 ( T ) ≥ … ) {\displaystyle {\big (}\sigma _{1}(T)\geq \sigma _{2}(T)\geq \dots {\big )}} . The largest singular value σ 1 ( T ) {\displaystyle \sigma _{1}(T)} is equal to the operator norm of T {\displaystyle T} (see Min-max theorem).
If T {\displaystyle T} acts on a Euclidean space (for example, R n {\displaystyle \mathbb {R} ^{n}} ), there is a simple geometric interpretation for the singular values: Consider the image by T {\displaystyle T} of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of T {\displaystyle T} (the figure provides an example in R 2 {\displaystyle \mathbb {R} ^{2}} ). If A {\displaystyle A} is a normal matrix, then its singular values are the absolute values of its eigenvalues. Indeed, in this case, the spectral theorem can be applied to obtain unitary diagonalization of A {\displaystyle A} as A = U Λ U ∗ {\displaystyle A=U\varLambda \,U^{*}} . Then, by definition, A ∗ A = U Λ ∗ Λ U ∗ = U | Λ | U ∗ {\displaystyle \textstyle {\sqrt {A^{*}A}}=U{\sqrt {\varLambda ^{*}\varLambda }}~U^{*}=U\,\vert \varLambda \vert ~U^{*}} . Most norms on Hilbert space operators studied are defined using singular values. For example, the Ky Fan k {\displaystyle k} -norm is the sum of the first k {\displaystyle k} singular values, the trace norm is the sum of all singular values, and the Schatten norm is the p {\displaystyle p} -th root of the sum of the p {\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 U Σ V ∗ {\displaystyle U\varSigma \,V^{*}} , where U {\displaystyle U} and V ∗ {\displaystyle V^{*}} are unitary matrices and Σ {\displaystyle \varSigma } is a rectangular diagonal matrix with the singular values lying on its diagonal. This is the singular value decomposition.
Basic properties For A ∈ C m × n {\displaystyle A\in \mathbb {C} ^{m\times n}} and i = 1 , 2 , … , min { m , n } {\displaystyle i=1,2,\ldots ,\min\{m,n\}} : Min-max theorem for singular values:Here, U {\displaystyle U} is a subspace of C n {\displaystyle \mathbb {C} ^{n}} ;
… excerpt ends here. Continue reading the full article.
