Spectral theorem
In linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix can be diagonalized. 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 finite-dimensional vector 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 modeled 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 decomposition, of the underlying vector space on which the operator acts.
Augustin-Louis Cauchy proved the spectral theorem for symmetric matrices, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about determinants. 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.
Finite-dimensional case
Hermitian maps and Hermitian matrices
We begin by considering a Hermitian matrix onThis condition implies that all eigenvalues of a Hermitian map are real: To see this, it is enough to apply it to the case when is an eigenvector.
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, there is at least one complex eigenvalue and corresponding eigenvector, which must by definition be non-zero. Then since
we find that is real. Now consider the space, the orthogonal complement of. By Hermiticity, is an invariant subspace of. To see that, consider any so that by definition of. To satisfy invariance, we need to check if. This is true because,. Applying the same argument to shows that has at least one real eigenvalue and corresponding eigenvector. This can be used to build another invariant subspace. Finite induction then finishes the proof.
The matrix representation of in a basis of eigenvectors is diagonal, and by the construction the proof gives a basis of mutually orthogonal eigenvectors; by choosing them to be unit vectors one obtains an orthonormal basis of eigenvectors. can be written as a linear combination of pairwise orthogonal projections, called its spectral decomposition. Let
be the eigenspace corresponding to an eigenvalue. Note that the definition does not depend on any choice of specific eigenvectors. In general, is the orthogonal direct sum of the spaces where the ranges over the spectrum of.
When the matrix being decomposed is Hermitian, the spectral decomposition is a special case of the Schur decomposition.
Spectral decomposition and the singular value decomposition
The spectral decomposition is a special case of the singular value decomposition, which states that any matrix can be expressed as, where and are unitary matrices and is a diagonal matrix. The diagonal entries of are uniquely determined by and are known as the singular values of. If is Hermitian, then and which implies.
Normal matrices
The spectral theorem extends to a more general class of matrices. Let be an operator on a finite-dimensional inner product space. is said to be normal if.One can show that is normal if and only if it is unitarily diagonalizable using the Schur decomposition. That is, any matrix can be written as, where is unitary and is upper triangular.
If is normal, then one sees that. Therefore, must be diagonal since a normal upper triangular matrix is diagonal. The converse is obvious.
In other words, is normal if and only if there exists a unitary matrix such that
where is a diagonal matrix. Then, the entries of the diagonal of are the eigenvalues of. The column vectors of are the eigenvectors of and they are orthonormal. Unlike the Hermitian case, the entries of need not be real.
Compact self-adjoint operators
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.As for Hermitian matrices, the key point is to prove the existence of at least one nonzero eigenvector. One cannot rely on determinants to show existence of eigenvalues, but one can use a maximization argument analogous to the variational characterization of eigenvalues.
If the compactness assumption is removed, then it is not true that every self-adjoint operator has eigenvectors. For example, the multiplication operator on which takes each to is bounded and self-adjoint, but has no eigenvectors. However, its spectrum, suitably defined, is still equal to, see spectrum of bounded operator.
Bounded self-adjoint operators
Possible absence of eigenvectors
The next generalization we consider is that of bounded self-adjoint operators on a Hilbert space. Such operators may have no eigenvectors: for instance let be the operator of multiplication by on, that is,This operator does not have any eigenvectors in, though it does have eigenvectors in a larger space. Namely the distribution, where is the Dirac delta function, is an eigenvector when construed in an appropriate sense. The Dirac delta function is however not a function in the classical sense and does not lie in the Hilbert space. Thus, the delta-functions are "generalized eigenvectors" of but not eigenvectors in the usual sense.
Spectral subspaces and projection-valued measures
In the absence of eigenvectors, one can look for a "spectral subspace" consisting of an almost eigenvector, i.e, a closed subspace of associated with a Borel set in the spectrum of. This subspace can be thought of as the closed span of generalized eigenvectors for with eigenvalues in. In the above example, where we might consider the subspace of functions supported on a small interval inside. This space is invariant under and for any in this subspace, is very close to. Each subspace, in turn, is encoded by the associated projection operator, and the collection of all the subspaces is then represented by a projection-valued measure.One formulation of the spectral theorem expresses the operator as an integral of the coordinate function over the operator's spectrum with respect to a projection-valued measure.
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 operator version
An alternative formulation of the spectral theorem says that every bounded self-adjoint operator is unitarily equivalent to a multiplication operator, a relatively simple type of operator.Multiplication operators are a direct generalization of diagonal matrices. A finite-dimensional Hermitian vector space may be coordinatized as the space of functions from a basis to the complex numbers, so that the -coordinates of a vector are the values of the corresponding function. The finite-dimensional spectral theorem for a self-adjoint operator states that there exists an orthonormal basis of eigenvectors, so that the inner product becomes the dot product with respect to the -coordinates: thus is isomorphic to for the discrete unit measure on. Also is unitarily equivalent to the multiplication operator, where is the eigenvalue of : that is, multiplies each -coordinate by the corresponding eigenvalue, the action of a diagonal matrix. Finally, the operator norm is equal to the magnitude of the largest eigenvector.
The spectral theorem is the beginning of the vast research area of functional analysis called operator theory; see also spectral measure.
There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. The only difference in the conclusion is that now may be complex-valued.
Direct integrals
There is also a formulation of the spectral theorem in terms of direct integrals. It is similar to the multiplication-operator formulation, but more canonical.Let be a bounded self-adjoint operator and let be the spectrum of. The direct-integral formulation of the spectral theorem associates two quantities to. First, a measure on, and second, a family of Hilbert spaces We then form the direct integral Hilbert space
The elements of this space are functions such that for all.
The direct-integral version of the spectral theorem may be expressed as follows:
The spaces can be thought of as something like "eigenspaces" for. Note, however, that unless the one-element set has positive measure, the space is not actually a subspace of the direct integral. Thus, the 's should be thought of as "generalized eigenspace"—that is, the elements of are "eigenvectors" that do not actually belong to the Hilbert space.
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 is canonical. Second, the function we are multiplying by is canonical in the direct-integral approach: Simply the function.