spectral theorem

Let U be a finite-dimensional, unitary space and let M:UU be an endomorphismPlanetmathPlanetmath. We say that M is normal if it commutes with its Hermitian adjoint, i.e.


Spectral Theorem

Let M:UU be a linear transformation of a unitary space. TFAE

  1. 1.

    The transformationMathworldPlanetmathPlanetmath M is normal.

  2. 2.


    Λ={λM-λE is singular},

    where E is the identity operatorMathworldPlanetmath, denote the spectrumPlanetmathPlanetmath (set of eigenvaluesMathworldPlanetmathPlanetmathPlanetmathPlanetmath) of M, the corresponding eigenspacesMathworldPlanetmath


    give an orthogonalMathworldPlanetmathPlanetmathPlanetmathPlanetmath, direct sumMathworldPlanetmathPlanetmath decomposition of U, i.e.


    and Eλ1Eλ2 for distinct eigenvalues λ1λ2.

  3. 3.

    We can decompose M as the sum


    where Λ is a finite subset of complex numbers indexing a family of commuting orthogonal projectionsPlanetmathPlanetmath Pλ:UU, i.e.

    Pλ=Pλ  PλPμ={Pλλ=μ0λμ,

    and where WLOG

  4. 4.

    There exists an orthonormal basisMathworldPlanetmath of U that diagonalizes M.


  1. 1.

    Here are some important classes of normal operators, distinguished by the nature of their eigenvalues.

  2. 2.

    There is a well-known version of the spectral theoremMathworldPlanetmath for , namely that a self-adjoint (symmetricMathworldPlanetmathPlanetmathPlanetmathPlanetmath) transformation of a real inner product spacesMathworldPlanetmath can diagonalized and that eigenvectorsMathworldPlanetmathPlanetmathPlanetmath corresponding to different eigenvalues are orthogonal. An even more down-to-earth version of this theorem says that a symmetric, real matrix can always be diagonalized by an orthonormal basis of eigenvectors.

  3. 3.

    There are several versions of increasing sophistication of the spectral theorem that hold in infinite-dimensional, Hilbert spaceMathworldPlanetmath setting. In such a context one must distinguish between the so-called discrete and continuousPlanetmathPlanetmath (no corresponding eigenspace) spectrums, and replace the representing sum for the operatorMathworldPlanetmath with some kind of an integral. The definition of self-adjointness is also quite tricky for unbounded operators. Finally, there are versions of the spectral theorem, of importance in theoretical quantum mechanics, that can be applied to continuous 1-parameter groups of commuting, self-adjoint operators.

Title spectral theorem
Canonical name SpectralTheorem
Date of creation 2013-03-22 12:45:49
Last modified on 2013-03-22 12:45:49
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 9
Author rmilson (146)
Entry type Theorem
Classification msc 15A23
Classification msc 15A63
Classification msc 15A18
Related topic DiagonalizableOperator
Defines normal operator