maths.free › Functional Analysis › Hilbert spaces and the spectral theorem

Hilbert spaces and the spectral theorem

Inner products, orthonormal bases, Fourier series, and eigenvalues of operators.

An inner product gives angles and projections; an orthonormal basis expands any vector: Fourier series are exactly this in L². Self-adjoint operators have real spectrum and an eigen-decomposition, the infinite-dimensional version of diagonalising a symmetric matrix. Picture it: the symmetric matrix [[2,1],[1,2]] stretching along two perpendicular eigenvectors. Think it: quantum observables are self-adjoint operators; measurement outcomes are their spectrum.

Ohatra · eigenvalues of [[2,1],[1,2]]

Linear-Algebra · Eigenvalues

Eigenvalues of [[2,1],[1,2]]

\left[\begin{matrix}2 & 1\\1 & 2\end{matrix}\right]

Dingana amin'ny dingana

  1. \det(A - \lambda I) = 0

    Eigenvalues are the roots of the characteristic polynomial.

  2. \det\left[\begin{matrix}2 - \lambda & 1\\1 & 2 - \lambda\end{matrix}\right] = 0

    Subtract λ from the diagonal.

  3. \lambda^{2} - 4 \lambda + 3 = 0

    Expand the determinant.

  4. \left(\lambda - 3\right) \left(\lambda - 1\right) = 0

    Factor.

  5. \lambda = 3, \lambda = 1

    Eigenvalues (with multiplicity).

  6. \lambda = 1:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}-1\\1\end{matrix}\right]

    Solve (A − 1I)v = 0 for a basis eigenvector.

  7. \lambda = 3:\ (A - \lambda I)\mathbf{v} = 0 \Rightarrow \mathbf{v} = \left[\begin{matrix}1\\1\end{matrix}\right]

    Solve (A − 3I)v = 0 for a basis eigenvector.

Asehoy ny valinteny
\lambda = 3,\; \lambda = 1

Ankehitriny ianao Misafidiana olana iray, na soraty na esory ny anao manokana. Ny dingana tsirairay, sary iray, ny valiny miafina mandra-pahafantaranao.

Aoka ho miasa ny anao

Ny kaonty maimaimpoana dia mampiditra ny fanamarihana amin'ny lesona tsirairay, ny tahirin'ny zavatra vitanao, ny olana voavahao ao amin'ny toerana iray, ary ny mpampianatra iray izay azonao anontanio momba ity pejy ity. misokatra ho an'ny rehetra ny matematika, na efa niditra na tsy niditra.

Misoratra anarana Fidirana

Marika ampiasaina eto

Tsindrio ny marika iray mba hahitana ny famaritana feno, sary, ary ny dikan'ny litera tsirairay ao aminy.

Ahoana ny fomba: Hilbert spaces and the spectral theorem

  1. Eigenvalues are the roots of the characteristic polynomial.
  2. Subtract λ from the diagonal.
  3. Expand the determinant.
  4. Factor.
  5. Eigenvalues (with multiplicity).
  6. Solve (A − 1I)v = 0 for a basis eigenvector.
  7. Solve (A − 3I)v = 0 for a basis eigenvector.

Fanontaniana napetrak'ireo olona

What is a Hilbert space?

A vector space with an inner product (so lengths and angles make sense) that is complete (no missing limit points). Square-integrable functions form one; quantum states live in one.

Mbola maro ao Functional Analysis

Normed and Banach spaces