In
mathematics, particularly
linear algebra and
functional analysis, a spectral theorem is a result about when a
linear operator or
matrix can be
diagonalized
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P and a diagonal matrix D such that or equivalently (Such D are not unique.) F ...
(that is, represented as a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
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 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 operator
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to its ...
s or more generally
normal operator
In mathematics, especially functional analysis, a normal operator on a complex Hilbert space ''H'' is a continuous linear operator ''N'' : ''H'' → ''H'' that commutes with its hermitian adjoint ''N*'', that is: ''NN*'' = ''N*N''.
Normal opera ...
s on
Hilbert spaces.
The spectral theorem also provides a
canonical
The adjective canonical is applied in many contexts to mean "according to the canon" the standard, rule or primary source that is accepted as authoritative for the body of knowledge or literature in that context. In mathematics, "canonical examp ...
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
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...
, 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
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -t ...
on
(but the following discussion will be adaptable to the more restrictive case of
symmetric matrices
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...
on
). We consider a
Hermitian map on a finite-dimensional
complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
inner product space endowed with a
positive definite sesquilinear inner product . The Hermitian condition on
means that for all ,
:
An equivalent condition is that , where is the
Hermitian conjugate
In mathematics, specifically in operator theory, each linear operator A on a Euclidean vector space defines a Hermitian adjoint (or adjoint) operator A^* on that space according to the rule
:\langle Ax,y \rangle = \langle x,A^*y \rangle,
where ...
of . In the case that is identified with a Hermitian matrix, the matrix of can be identified with its
conjugate transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \boldsymbol is an n \times m matrix obtained by transposing \boldsymbol and applying complex conjugate on each entry (the complex c ...
. (If is a
real matrix, then this is equivalent to , that is, is a
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with ...
.)
This condition implies that all eigenvalues of a Hermitian map are real: it is enough to apply it to the case when is an eigenvector. (Recall that an
eigenvector
In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted ...
of a linear map is a (non-zero) vector such that for some scalar . The value is the corresponding
eigenvalue. Moreover, the
eigenvalues are roots of the
characteristic polynomial.)
Theorem. If is Hermitian on , then there exists an
orthonormal basis of consisting of eigenvectors of . Each eigenvalue is real.
We provide a sketch of a proof for the case where the underlying field of scalars is the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s.
By the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
, applied to the
characteristic polynomial of , there is at least one eigenvalue and eigenvector . Then since
:
we find that is real. Now consider the space , the
orthogonal complement In the mathematical fields of linear algebra and functional analysis, the orthogonal complement of a subspace ''W'' of a vector space ''V'' equipped with a bilinear form ''B'' is the set ''W''⊥ of all vectors in ''V'' that are orthogonal to every ...
of . By Hermiticity, is an
invariant subspace of . Applying the same argument to shows that has an eigenvector . Finite induction then finishes the proof.
The spectral theorem holds also for symmetric maps on finite-dimensional real inner product spaces, but the existence of an eigenvector does not follow immediately from the
fundamental theorem of algebra
The fundamental theorem of algebra, also known as d'Alembert's theorem, or the d'Alembert–Gauss theorem, states that every non- constant single-variable polynomial with complex coefficients has at least one complex root. This includes polynomia ...
. To prove this, consider as a Hermitian matrix and use the fact that all eigenvalues of a Hermitian matrix are real.
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. is the orthogonal direct sum of the spaces where the index ranges over eigenvalues.
In other words, if denotes the
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
onto , and are the eigenvalues of , 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 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 be an operator on a finite-dimensional inner product space. is said to be
normal Normal(s) or The Normal(s) may refer to:
Film and television
* ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson
* ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie
* ''Norma ...
if . One can show that is normal if and only if it is unitarily diagonalizable. Proof: By the
Schur decomposition, we can write any matrix 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 (see
normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose :
The concept of normal matrices can be extended to normal operators on infinite dimensional normed spaces and to normal elements in C*-algebras. As ...
). The converse is obvious.
In other words, is normal if and only if there exists a
unitary matrix
In linear algebra, a complex square matrix is unitary if its conjugate transpose is also its inverse, that is, if
U^* U = UU^* = UU^ = I,
where is the identity matrix.
In physics, especially in quantum mechanics, the conjugate transpose is ...
such that
:
where is a
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal m ...
. 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
In mathematics, a self-adjoint operator on an infinite-dimensional complex vector space ''V'' with inner product \langle\cdot,\cdot\rangle (equivalently, a Hermitian operator in the finite-dimensional case) is a linear map ''A'' (from ''V'' to itse ...
is virtually the same as in the finite-dimensional case.
Theorem. Suppose is a compact self-adjoint operator on a (real or complex) Hilbert space . Then there is an
orthonormal basis of consisting of eigenvectors of . 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, then it is ''not'' true that every self-adjoint operator has eigenvectors.
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 eigenvalues: 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 or any other
Banach 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 (true) eigenvectors, one can look for subspaces consisting of ''almost eigenvectors''. In the above example, for example, where
we might consider the subspace of functions supported on a small interval