Singular Value Decomposition - Existence

Existence

An eigenvalue λ of a matrix is characterized by the algebraic relation M u = λ u. When M is Hermitian, a variational characterization is also available. Let M be a real n × n symmetric matrix. Define f :Rn → R by f(x) = xT M x. By the extreme value theorem, this continuous function attains a maximum at some u when restricted to the closed unit sphere {||x|| ≤ 1}. By the Lagrange multipliers theorem, u necessarily satisfies

where the nabla symbol, is the del operator.

A short calculation shows the above leads to M u = λ u (symmetry of M is needed here). Therefore λ is the largest eigenvalue of M. The same calculation performed on the orthogonal complement of u gives the next largest eigenvalue and so on. The complex Hermitian case is similar; there f(x) = x* M x is a real-valued function of 2n real variables.

Singular values are similar in that they can be described algebraically or from variational principles. Although, unlike the eigenvalue case, Hermiticity, or symmetry, of M is no longer required.

This section gives these two arguments for existence of singular value decomposition.