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:
“I create my social existence by earning and spending.”
—Mason Cooley (b. 1927)
“Suppose that humans happen to be so constructed that they desire the opportunity for freely undertaken productive work. Suppose that they want to be free from the meddling of technocrats and commissars, bankers and tycoons, mad bombers who engage in psychological tests of will with peasants defending their homes, behavioral scientists who can’t tell a pigeon from a poet, or anyone else who tries to wish freedom and dignity out of existence or beat them into oblivion.”
—Noam Chomsky (b. 1928)
“Each of us, even the lowliest and most insignificant among us, was uprooted from his innermost existence by the almost constant volcanic upheavals visited upon our European soil and, as one of countless human beings, I can’t claim any special place for myself except that, as an Austrian, a Jew, writer, humanist and pacifist, I have always been precisely in those places where the effects of the thrusts were most violent.”
—Stefan Zweig (18811942)