Singular value

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In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator T acting on a Hilbert space are defined as the eigenvalues of the operator $\sqrt{T^*T}$ (where T^* denotes the adjoint of T and the square root is taken in the operator sense). The singular values are nonnegative real numbers, usually listed in decreasing order s₁(T), s₂(T), ... . The largest singular value s₁(T) is equal to the operator norm of T. In the case of a normal matrix A, the spectral theorem can be applied to obtain unitary diagonalization of A as per $A = U Λ U *$ . Therefore, $\sqrt{A^*A}=U|\Lambda|U^*$ and so the singular values are simply the absolute values of the eigenvalues.

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, Allakhverdiev proved the following characterization of the nth s-number:

$s_n(T)=\inf\{\, \|T-L\| : L\ \mbox{is}\ \mbox{an}\ \mbox{operator}\ \mbox{of}\ \mbox{finite}\ \mbox{rank}\ <n\, \}.$

This formulation made it possible to extend the notion of s-numbers to operators in Banach space.

Most 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 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 s-numbers are useful in classifying different operators.

In the finite-dimensional case, a matrix can always be decomposed in the form UDW, where U and W are unitary matrices and D is a diagonal matrix with the singular values lying on the diagonal. This is the singular value decomposition.

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