Spectral theorem

In <a href="/facts/Linear_algebra/8VaB4VWk">linear algebra</a> and <a href="/facts/Functional_analysis/MIp233MT">functional analysis</a>, a spectral theorem is a result about when a <a href="/facts/Linear_operator/Q1PBKNVV">linear operator</a> or <a href="/facts/Matrix_(mathematics)/qa8FI4ko">matrix</a> can be <a href="/facts/Diagonalizable_matrix/y3MkyTCy">diagonalized</a> (that is, represented as a <a href="/facts/Diagonal_matrix/tjIAHrE6">diagonal matrix</a> in some basis). This is extremely useful because computations involving a diagonalizable matrix can often be reduced to much simpler computations involving the corresponding diagonal matrix. The concept of diagonalization is relatively straightforward for operators on <a href="/facts/Finite-dimensional_vector_space/z8m02A3W">finite-dimensional vector spaces</a> but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of <a href="/facts/Linear_operator/Q1PBKNVV">linear operators</a> that can be modeled by <a href="/facts/Multiplication_operator/YrvjuoVQ">multiplication operators</a>, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative <a href="/facts/C*-algebra/B3tnGHIo">C*-algebras</a>. See also <a href="/facts/Spectral_theory/8sAEEIMM">spectral theory</a> for a historical perspective.
Examples of operators to which the spectral theorem applies are <a href="/facts/Self-adjoint_operator/VrH1nEAK">self-adjoint operators</a> or more generally <a href="/facts/Normal_operator/lwtz7tuF">normal operators</a> on <a href="/facts/Hilbert_space/aDFMh4lV">Hilbert spaces</a>.
The spectral theorem also provides a <a href="/facts/Canonical_form/K2jhB44k">canonical</a> decomposition, called the <a href="/facts/Eigendecomposition_of_a_matrix/a2nfF7hJ">spectral decomposition</a>, of the underlying vector space on which the operator acts.
<a href="/facts/Augustin-Louis_Cauchy/aoTjgj6j">Augustin-Louis Cauchy</a> proved the spectral theorem for <a href="/facts/Symmetric_matrix/08CGwjNl">symmetric matrices</a>, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about <a href="/facts/Determinant/eGYqJ2TF">determinants</a>. The spectral theorem as generalized by <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a> is today perhaps the most important result of <a href="/facts/Operator_theory/Ays1ofel">operator theory</a>.
This article mainly focuses on the simplest kind of spectral theorem, that for a <a href="/facts/Self-adjoint/rKqGtlpy">self-adjoint</a> operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.

Spectral theorem open-in-new

Spectral theorem