Tag Archives: theorem

Spectral theorem

Spectral theorem

From Wikipedia, the free encyclopedia
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can bediagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.
Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.
The spectral theorem also provides a canonical decomposition, called the spectral decompositioneigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.
In this article we consider mainly the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.

Best approximation theorem

Theorem

Let X be an inner product space with induced norm, and A\subseteq X a non-emptycomplete convex subset. Then, for all x\in X, there exists a unique best approximation a0 to x in A.

Proof

Suppose x = 0 (if not the case, consider A − {x} instead) and let d=d(0,A)=\inf_{a\in A} ||a||. There exists a sequence (an) in Asuch that
  • ||a_n||\to d.
We now prove that (an) is a Cauchy sequence. By the parallelogram rule, we get
  • ||\frac{a_n-a_m}{2}||^2+||\frac{a_n+a_m}{2}||^2=\frac{1}{2}(||a_n||^2+||a_m||^2).
Since A is convex\frac{a_n+a_m}{2}\in A so
  • \underset{m,n\in \mathbb N}{\forall }\; ||\frac{a_n+a_m}{2}||\geq d.
Hence
  • ||\frac{a_n-a_m}{2}||^2\leq \frac{1}{2}(||a_n||^2+||a_m||^2)-d^2\to 0 as m,n\to \infty
which implies ||a_n-a_m||\to 0 as m,n\to \infty. In other words, (an) is a Cauchy sequence. Since A is complete,
  • \underset{a_0\in A}{\exists }\; a_n\to a_0.
Since a_0\in A||a_0||\geq d. Furthermore
  • ||a_0||\leq ||a_0-a_n||+||a_n||\to d as n\to \infty,
which proves | | a0 | | = d. Existence is thus proved. We now prove uniqueness. Suppose there were two distinct best approximations a0and a0 to x (which would imply | | a0 | | = | | a0‘ | | = d). By the parallelogram rule we would have
  • ||\frac{a_0+a_0'}{2}||^2+||\frac{a_0-a_0'}{2}||^2=\frac{1}{2}(||a_0||^2+||a_0'||^2)=d^2.
Then
  • ||\frac{a_0+a_0'}{2}||^2<d^2
which cannot happen since A is convex, and as such \frac{a_0+a_0'}{2}\in A, which means ||\frac{a_0+a_0'}{2}||^2\geq d^2, thus completing the proof.

Ehrenfest theorem

The Ehrenfest theorem, named after Paul Ehrenfest, the Austrian physicist and mathematician, relates the time derivative of the expectation value for a quantum mechanicaloperator to the commutator of that operator with the Hamiltonian of the system. It is

\frac{d}{dt}\langle A\rangle = \frac{1}{i\hbar}\langle [A,H] \rangle+ \left\langle \frac{\partial A}{\partial t}\right\rangle

where A is some QM operator and \langle A\rangle is its expectation value. Ehrenfest’s theorem is obvious in the Heisenberg picture of quantum mechanics, where it is just the expectation value of the Heisenberg equation of motion.

Ehrenfest’s theorem is closely related to Liouville’s theorem from Hamiltonian mechanics, which involves the Poisson bracket instead of a commutator. In fact, it is a rule of thumb that a theorem in quantum mechanics which contains a commutator can be turned into a theorem in classical mechanics by changing the commutator into a Poisson bracket and multiplying by i\hbar.

http://en.wikipedia.org/wiki/Ehrenfest_theorem

Noether’s (first) theorem

Noether’s (first) theorem states that any differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proved by German mathematician Emmy Noether in 1915 and published in 1918.[1] The action of a physical system is the integral over time of a Lagrangian function (which may or may not be an integral over space of a Lagrangian density function), from which the system’s behavior can be determined by the principle of least action.

Noether’s theorem has become a fundamental tool of modern theoretical physics and the calculus of variations. A generalization of the seminal formulations on constants of motion in Lagrangian and Hamiltonian mechanics (1788 and 1833, respectively), it does not apply to systems that cannot be modeled with a Lagrangian; for example, dissipative systems with continuous symmetries need not have a corresponding conservation law.

For illustration, if a physical system behaves the same regardless of how it is oriented in space, its Lagrangian is rotationally symmetric; from this symmetry, Noether’s theorem shows the angular momentum of the system must be conserved. The physical system itself need not be symmetric; a jagged asteroid tumbling in space conserves angular momentum despite its asymmetry – it is the laws of motion that are symmetric. As another example, if a physical experiment has the same outcome regardless of place or time (having the same outcome, say, somewhere in Asia on a Tuesday or in America on a Wednesday), then its Lagrangian is symmetric under continuous translations in space and time; by Noether’s theorem, these symmetries account for the conservation laws of linear momentum and energy within this system, respectively. (These examples are just for illustration; in the first one, Noether’s theorem added nothing new – the results were known to follow from Lagrange’s equations and from Hamilton’s equations.)

Noether’s theorem is important, both because of the insight it gives into conservation laws, and also as a practical calculational tool. It allows researchers to determine the conserved quantities from the observed symmetries of a physical system. Conversely, it allows researchers to consider whole classes of hypothetical Lagrangians to describe a physical system. For illustration, suppose that a new field is discovered that conserves a quantity X. Using Noether’s theorem, the types of Lagrangians that conserve X because of a continuous symmetry can be determined, and then their fitness judged by other criteria.

There are numerous different versions of Noether’s theorem, with varying degrees of generality. The original version only applied to ordinary differential equations (particles) and not partial differential equations (fields). The original versions also assume that the Lagrangian only depends upon the first derivative, while later versions generalize the theorem to Lagrangians depending on the nth derivative. There is also a quantum version of this theorem, known as the Ward–Takahashi identity. Generalizations of Noether’s theorem to superspaces also exist.

http://en.wikipedia.org/wiki/Noether’s_theorem

Symmetry in Science: An Introduction to the General Theory

http://www.lightandmatter.com/bk2a.pdf

http://mathworld.wolfram.com/Symmetry.html

 

 

 

Best approximation theorem

Best approximation theorem

Theorem

Let X be an inner product space with induced norm, and Asubseteq X a non-emptycomplete convex subset. Then, for all xin X, there exists a unique best approximation a0 to x in A.

Proof

Suppose x = 0 (if not the case, consider A − {x} instead) and let d=d(0,A)=inf_{ain A} ||a||. There exists a sequence (an) in Asuch that
  • ||a_n||to d.
We now prove that (an) is a Cauchy sequence. By the parallelogram rule, we get
  • ||frac{a_n-a_m}{2}||^2+||frac{a_n+a_m}{2}||^2=frac{1}{2}(||a_n||^2+||a_m||^2).
Since A is convexfrac{a_n+a_m}{2}in A so
  • underset{m,nin mathbb N}{forall }; ||frac{a_n+a_m}{2}||geq d.
Hence
  • ||frac{a_n-a_m}{2}||^2leq frac{1}{2}(||a_n||^2+||a_m||^2)-d^2to 0 as m,nto infty
which implies ||a_n-a_m||to 0 as m,nto infty. In other words, (an) is a Cauchy sequence. Since A is complete,
  • underset{a_0in A}{exists }; a_nto a_0.
Since a_0in A||a_0||geq d. Furthermore
  • ||a_0||leq ||a_0-a_n||+||a_n||to d as nto infty,
which proves | | a0 | | = d. Existence is thus proved. We now prove uniqueness. Suppose there were two distinct best approximations a0and a0 to x (which would imply | | a0 | | = | | a0‘ | | = d). By the parallelogram rule we would have
  • ||frac{a_0+a_0'}{2}||^2+||frac{a_0-a_0'}{2}||^2=frac{1}{2}(||a_0||^2+||a_0'||^2)=d^2.
Then
  • <img alt="||frac{a_0+a_0'}{2}||^2
which cannot happen since A is convex, and as such frac{a_0+a_0'}{2}in A, which means ||frac{a_0+a_0'}{2}||^2geq d^2, thus completing the proof.

Spectral theorem

Spectral theorem

From Wikipedia, the free encyclopedia
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can bediagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional spaces, but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of linear operators that can be modelled by multiplication operators, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative C*-algebras. See also spectral theory for a historical perspective.
Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces.
The spectral theorem also provides a canonical decomposition, called the spectral decompositioneigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.
In this article we consider mainly the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.