Spectral theorem
In mathematics, particularly 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). 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 finitedimensional vector spaces but requires some modification for operators on infinitedimensional 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 selfadjoint operators or more generally normal operators on Hilbert spaces.
The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which the operator acts.
AugustinLouis Cauchy proved the spectral theorem for selfadjoint matrices, i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about determinants.^{[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 selfadjoint operator on a Hilbert space. However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.
Contents
Finitedimensional case
Hermitian maps and Hermitian matrices
We begin by considering a Hermitian matrix on (but the following discussion will be adaptable to the more restrictive case of symmetric matrices on ). We consider a Hermitian map A on a finitedimensional complex inner product space V endowed with a positive definite sesquilinear inner product . The Hermitian condition on means that for all x, y ∈ V,
(An equivalent condition is that A^{∗} = A, where A^{∗} is the hermitian conjugate of A.) In the case that A is identified with a Hermitian matrix, the matrix of A^{∗} can be identified with its conjugate transpose. (If A is a real matrix, this is equivalent to A^{T} = A, that is, A is a symmetric matrix.)
This condition implies that all eigenvalues of a Hermitian map are real: 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 (nonzero) vector x such that Ax = λx for some scalar λ. The value λ is the corresponding eigenvalue. Moreover, the eigenvalues are solutions to the characteristic polynomial.)
Theorem. If A is Hermitian, there exists an orthonormal basis of V consisting of eigenvectors of A. Each eigenvalue is real.
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 eigenvalue λ_{1} and eigenvector e_{1}. Then since
we find that λ_{1} is real. Now consider the space K = span{e_{1}}^{⊥}, the orthogonal complement of e_{1}. By Hermiticity, K is an invariant subspace of A. Applying the same argument to K shows that A has an eigenvector e_{2} ∈ K. Finite induction then finishes the proof.
The spectral theorem holds also for symmetric maps on finitedimensional real inner product spaces, but the existence of an eigenvector does not follow immediately from the fundamental theorem of algebra. To prove this, consider A as a Hermitian matrix and use the fact that all eigenvalues of a Hermitian matrix are real.
If one chooses the eigenvectors of A as an orthonormal basis, the matrix representation of A in this basis is diagonal. Equivalently, A can be written as a linear combination of pairwise orthogonal projections, called its spectral decomposition. Let
be the eigenspace corresponding to an eigenvalue λ. Note that the definition does not depend on any choice of specific eigenvectors. V is the orthogonal direct sum of the spaces V_{λ} where the index ranges over eigenvalues.
In other words, if P_{λ} denotes the orthogonal projection onto V_{λ}, and λ_{1}, ..., λ_{m} are the eigenvalues of A, then the spectral decomposition may be written as
If the spectral decomposition of A is , then and for any scalar It follows that for any polynomial f one has
The spectral decomposition is a special case of both the Schur decomposition and the singular value decomposition.
Normal matrices
The spectral theorem extends to a more general class of matrices. Let A be an operator on a finitedimensional inner product space. A is said to be normal if A^{∗}A = AA^{∗}. One can show that A is normal if and only if it is unitarily diagonalizable. Proof: By the Schur decomposition, we can write any matrix as A = UTU^{∗}, where U is unitary and T is uppertriangular. If A is normal, one sees that TT^{∗} = T^{*}T. Therefore, T must be diagonal since a normal upper triangular matrix is diagonal (see normal matrix). The converse is obvious.
In other words, A is normal if and only if there exists a unitary matrix U such that
where D is a diagonal matrix. Then, the entries of the diagonal of D are the eigenvalues of A. The column vectors of U are the eigenvectors of A and they are orthonormal. Unlike the Hermitian case, the entries of D need not be real.
Compact selfadjoint operators
In the more general setting of Hilbert spaces, which may have an infinite dimension, the statement of the spectral theorem for compact selfadjoint operators is virtually the same as in the finitedimensional case.
Theorem. Suppose A is a compact selfadjoint operator on a (real or complex) Hilbert space V. Then there is an orthonormal basis of V consisting of eigenvectors of A. Each eigenvalue is real.
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, it is not true that every selfadjoint operator has eigenvectors.
Bounded selfadjoint operators
Possible absence of eigenvectors
The next generalization we consider is that of bounded selfadjoint operators on a Hilbert space. Such operators may have no eigenvalues: for instance let A be the operator of multiplication by t on L^{2}[0, 1], that is,^{[3]}
Now, a physicist would say that does have eigenvectors, namely the , where is a Dirac deltafunction. A deltafunction, however, is not a normalizable function; that is, it is not actually in the Hilbert space L^{2}[0, 1]. Thus, the deltafunctions are "generalized eigenvectors" but not eigenvectors in the strict sense.
Spectral subspaces and projectionvalued measures
In the absence of (true) eigenvectors, one can look for subspaces consisting of almost eigenvectors. In the above example, for example, 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 . In this approach to the spectral theorem, if is a bounded selfadjoint operator, one looks for large families of such "spectral subspaces".^{[4]} Each subspace, in turn, is encoded by the associated projection operator, and the collection of all the subspaces is then represented by a projectionvalued measure.
One formulation of the spectral theorem expresses the operator A as an integral of the coordinate function over the operator's spectrum with respect to a projectionvalued measure.^{[5]}
When the selfadjoint operator in question is compact, this version of the spectral theorem reduces to something similar to the finitedimensional 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 selfadjoint operator is unitarily equivalent to a multiplication operator. The significance of this result is that multiplication operators are in many ways easy to understand.
Theorem.^{[6]} Let A be a bounded selfadjoint operator on a Hilbert space H. Then there is a measure space (X, Σ, μ) and a realvalued essentially bounded measurable function f on X and a unitary operator U:H → L^{2}_{μ}(X) such that
 where T is the multiplication operator:
 and
The spectral theorem is the beginning of the vast research area of functional analysis called operator theory; see also the 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 f may be complexvalued.
Direct integrals
There is also a formulation of the spectral theorem in terms of direct integrals. It is similar to the multiplicationoperator formulation, but more canonical.
Let be a bounded selfadjoint operator and let be the spectrum of . The directintegral 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 (or "sections") such that for all . The directintegral version of the spectral theorem may be expressed as follows:^{[7]}

Theorem. If is a bounded selfadjoint operator, then is unitarily equivalent to the "multiplication by " operator on
for some measure and some family of Hilbert spaces. The measure is uniquely determined by up to measuretheoretic equivalence; that is, any two measure associated to the same have the same sets of measure zero. The dimensions of the Hilbert spaces are uniquely determined by up to a set of measure zero.
The spaces can be thought of as something like "eigenspaces" for . Note, however, that unless the oneelement 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 multiplicationoperator and direct integral formulations of the spectral theorem express a selfadjoint 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. Second, the function we are multiplying by is canonical in the directintegral approach: Simply the function .
Cyclic vectors and simple spectrum
A vector is called a cyclic vector for if the vectors span a dense subspace of the Hilbert space. Suppose is a bounded selfadjoint operator for which a cyclic vector exists. In that case, there is no distinction between the directintegral and multiplicationoperator formulations of the spectral theorem. Indeed, in that case, there is a measure on the spectrum of such that is unitarily equivalent to the "multiplication by " operator on .^{[8]} This result represents simultaneously a multiplication operator and as a direct integral, since is just a direct integral in which each Hilbert space is just .
Not every bounded selfadjoint operator admits a cyclic vector; indeed, by the uniqueness in the direct integral decomposition, this can occur only when all the 's have dimension one. When this happens, we say that has "simple spectrum" in the sense of spectral multiplicity theory. That is, a bounded selfadjoint operator that admits a cyclic vector should be thought of as the infinitedimensional generalization of a selfadjoint matrix with distinct eigenvalues (i.e., each eigenvalue has multiplicity one).
Although not every 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 has a cyclic vector. This observation is the key to the proofs of the multiplicationoperator and directintegral forms of the spectral theorem.
Functional calculus
One important application of the spectral theorem (in whatever form) is the idea of defining a functional calculus. That is, given a function defined on the spectrum of , we wish to define an operator . If is simply a positive power, , then is just the power of , . The interesting cases are where is a nonpolynomial function such as a square root or an exponential. Either of the versions of the spectral theorem provides such a functional calculus.^{[9]} In the directintegral version, for example, acts as the "multiplication by " operator in the direct integral:
 .
That is to say, each space in the direct integral is a (generalized) eigenspace for with eigenvalue .
General selfadjoint operators
Many important linear operators which occur in analysis, such as differential operators, are unbounded. There is also a spectral theorem for selfadjoint operators that applies in these cases. To give an example, every constantcoefficient differential operator is unitarily equivalent to a multiplication operator. Indeed, the unitary operator that implements this equivalence is the Fourier transform; the multiplication operator is a type of Fourier multiplier.
In general, spectral theorem for selfadjoint operators may take several equivalent forms.^{[10]} Notably, all of the formulations given in the previous section for bounded selfadjoint operators—the projectionvalued measure version, the multiplicationoperator version, and the directintegral version—continue to hold for unbounded selfadjoint operators, with small technical modifications to deal with domain issues.
See also
 Borel functional calculus
 Spectral theory
 Matrix decomposition
 Canonical form
 Jordan decomposition, of which the spectral decomposition is a special case.
 Singular value decomposition, a generalisation of spectral theorem to arbitrary matrices.
 Eigendecomposition of a matrix
Notes
 ^ Hawkins, Thomas (1975). "Cauchy and the spectral theory of matrices". Historia Mathematica. 2: 1–29. doi:10.1016/03150860(75)900324.
 ^ A Short History of Operator Theory by Evans M. Harrell II
 ^ Hall 2013 Section 6.1
 ^ Hall 2013 Theorem 7.2.1
 ^ Hall 2013 Theorem 7.12
 ^ Hall 2013 Theorem 7.20
 ^ Hall 2013 Theorem 7.19
 ^ Hall 2013 Lemma 8.11
 ^ E.g., Hall 2013 Definition 7.13
 ^ See Section 10.1 of Hall 2013
References
 Sheldon Axler, Linear Algebra Done Right, Springer Verlag, 1997
 Hall, B.C. (2013), Quantum Theory for Mathematicians, Graduate Texts in Mathematics, 267, Springer, ISBN 9781461471158
 Paul Halmos, "What Does the Spectral Theorem Say?", American Mathematical Monthly, volume 70, number 3 (1963), pages 241–247 Other link
 M. Reed and B. Simon, Methods of Mathematical Physics, vols I–IV, Academic Press 1972.
 G. Teschl, Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators, http://www.mat.univie.ac.at/~gerald/ftp/bookschroe/, American Mathematical Society, 2009.