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).
Visualisation of a singular value decomposition (SVD) of a 2-dimensional, real shearing matrixM. First, we see the unit disc in blue together with the two canonical unit vectors. We then see the action of M, which distorts the disc to an ellipse. The SVD decomposes M into three simple transformations: a rotationV*, a scalingΣ along the rotated coordinate axes and a second rotation U. Σ is a diagonal matrix containing in its diagonal the singular values of M, which represent the lengths σ1 and σ2 of the semi-axes of the ellipse.
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).
The singular values are the absolute values of the eigenvalues of a normal matrixA, because the spectral theorem can be applied to obtain unitary diagonalization of A as A = UΛU*. Therefore, .
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.
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, Allahverdiev proved the following characterization of the nth s-number [1]:
This formulation made it possible to extend the notion of s-numbers to operators in Banach space.
^I. C. Gohberg and M. G. Krein. Introduction to the Theory of Linear Non-selfadjoint Operators. American Mathematical Society, Providence, R.I.,1969. Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18.
^R.A. Horn and C.R. Johnson. Topics in Matrix Analysis. Cambridge University Press, Cambridge, 1991. Chap. 3