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.
Read more about this topic: Singular Value Decomposition
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—Norman Mailer (b. 1923)
“When I think of God, when I think of him as existent, and when I believe him to be existent, my idea of him neither increases nor diminishes. But as it is certain there is a great difference betwixt the simple conception of the existence of an object, and the belief of it, and as this difference lies not in the parts or composition of the idea which we conceive; it follows, that it must lie in the manner in which we conceive it.”
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