In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized (that is, represented as a diagonal matrix in some basis).
In more abstract language, the spectral theorem is a statement about commutative C*-algebras.
[1][2] The spectral theorem as generalized by John von Neumann is today perhaps the most important result of operator theory.
This article mainly focuses on 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.
This condition implies that all eigenvalues of a Hermitian map are real: To see this, it is enough to apply it to the case when x = y is an eigenvector.
(Recall that an eigenvector of a linear map A is a non-zero vector v such that A v = λv for some scalar λ.
Theorem — If A is Hermitian on V, then there exists an orthonormal basis of V consisting of eigenvectors of A.
We provide a sketch of a proof for the case where the underlying field of scalars is the complex numbers.
By the fundamental theorem of algebra, applied to the characteristic polynomial of A, there is at least one complex eigenvalue λ1 and corresponding eigenvector v1 , which must by definition be non-zero.
A can be written as a linear combination of pairwise orthogonal projections, called its spectral decomposition.
In the more general setting of Hilbert spaces, which may have an infinite dimension, the statement of the spectral theorem for compact self-adjoint operators is virtually the same as in the finite-dimensional case.
Theorem — Suppose A is a compact self-adjoint operator on a (real or complex) Hilbert space V. Then there is an orthonormal basis of V consisting of eigenvectors of A.
As for Hermitian matrices, the key point is to prove the existence of at least one nonzero eigenvector.
If the compactness assumption is removed, then it is not true that every self-adjoint operator has eigenvectors.
When the self-adjoint operator in question is compact, this version of the spectral theorem reduces to something similar to the finite-dimensional spectral theorem above, except that the operator is expressed as a finite or countably infinite linear combination of projections, that is, the measure consists only of atoms.
Multiplication operators are a direct generalization of diagonal matrices.
There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces.
Although both the multiplication-operator and direct integral formulations of the spectral theorem express a self-adjoint operator as unitarily equivalent to a multiplication operator, the direct integral approach is more canonical.
First, the set over which the direct integral takes place (the spectrum of the operator) is canonical.
In that case, there is no distinction between the direct-integral and multiplication-operator formulations of the spectral theorem.
Not every bounded self-adjoint operator admits a cyclic vector; indeed, by the uniqueness in the direct integral decomposition, this can occur only when all the
has "simple spectrum" in the sense of spectral multiplicity theory.
That is, a bounded self-adjoint operator that admits a cyclic vector should be thought of as the infinite-dimensional generalization of a self-adjoint matrix with distinct eigenvalues (i.e., each eigenvalue has multiplicity one).
admits a cyclic vector, it is easy to see that we can decompose the Hilbert space as a direct sum of invariant subspaces on which
This observation is the key to the proofs of the multiplication-operator and direct-integral forms of the spectral theorem.
One important application of the spectral theorem (in whatever form) is the idea of defining a functional calculus.
In general, spectral theorem for self-adjoint operators may take several equivalent forms.
[10] Notably, all of the formulations given in the previous section for bounded self-adjoint operators—the projection-valued measure version, the multiplication-operator version, and the direct-integral version—continue to hold for unbounded self-adjoint operators, with small technical modifications to deal with domain issues.
Though, for self-adjoint operators there always exist a real subset of "generalized eigenvalues" such that the corresponding set of eigenvectors is complete.