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
Famous quotes containing the word existence:
“No other group in America has so had their identity socialized out of existence as have black women.... When black people are talked about the focus tends to be on black men; and when women are talked about the focus tends to be on white women.”
—bell hooks (b. c. 1955)
“The individual who has to justify his existence by his own efforts is in eternal bondage to himself.”
—Eric Hoffer (1902–1983)
“During a walk or in a book or in the middle of an embrace, suddenly I awake to a stark amazement at everything. The bare fact of existence paralyzes me... To be alive is so incredible that all I can do is to lie still and merely breathe—like an infant on its back in a cot. It is impossible to be interested in anything in particular while overhead the sun shines or underneath my feet grows a single blade of grass.”
—W.N.P. Barbellion (1889–1919)