Singular value

In mathematics, in particular functional analysis, the singular values, or s-numbers of a compact operator T : X → Y acting between Hilbert spaces X and Y, are the square roots of the eigenvalues of the non-negative self-adjoint operator T*T : X → X (where T* denotes the adjoint of T).

The singular values are non-negative 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 (see Min-max theorem).

In the case that T acts on euclidean space Rⁿ, there is a simple geometric interpretation for the singular values: Consider the image by T of the unit sphere; this is an ellipsoid, and the lengths of its semi-axes are the singular values of T (the figure provides an example in R²).

In the case of a normal matrix A, the spectral theorem can be applied to obtain unitary diagonalization of A as per A = UΛU*. Therefore, ${\displaystyle {\sqrt {A^{*}A}}=U|\Lambda |U^{*}}$ and so the singular values are simply the absolute values of the eigenvalues.

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