Information about Spectral Decomposition
In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. In broad terms the spectral theorem provides conditions under which an operator or a matrix can be diagonalized (that is, represented as a diagonal matrix in some basis). This concept of diagonalization is relatively straightforward for operators on finite-dimensional 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 modelled by multiplication operators, which are as simple as one can hope to find. 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, or eigendecomposition, of the underlying vector space on which it acts.
In this article we consider mainly 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.
for all x, y elements of V.
An equivalent condition is that A* = A, where A* is the conjugate transpose of A. If A is a real matrix, this is also equivalent to AT = A
Recall that an eigenvector of a linear operator A is a (non-zero) vector x such that Ax = λx for some scalar λ. The value λ is the corresponding eigenvalue.
Theorem. There is an orthonormal basis of V consisting of eigenvectors of A. Each eigenvalue is real.
This result is of such importance in many parts of mathematics, that we provide a sketch of a proof for the case wherein the underlying field of scalars is the complex numbers. First we show that all the eigenvalues are real. Suppose that λ is an eigenvalue of A with corresponding eigenvector x. Thus
Since x is non-zero, it follows that λ equals its own conjugate and is therefore real.
To prove the existence of an eigenvector basis, we use induction on the dimension of V. In fact it suffices to show A has at least one non-zero eigenvector e. For then we can consider the space K of vectors v orthogonal to e. This is finite-dimensional because it is a subspace of a finite dimensional space, and A has the property that it maps every vector w in K into K. This is shown as follows: If w ∈ K, then using the symmetry property of A,
Moreover, A considered as a linear operator on K is also symmetric, so by the induction hypothesis there is a basis for V consisting of eigenvectors of A.
It remains, however, to show that A has at least one eigenvector. Since the ground field is algebraically closed, the polynomial function (called the characteristic polynomial of A)
has a complex root r. This implies the linear operator A − rI is not invertible and hence maps a non-zero vector e to 0. This vector e is a non-zero eigenvector of A. This implies that r is an eigenvalue, so is actually a real number. This completes the proof.
Notice the second part of the proof works for any square matrices. Clearly any square matrix has at least one eigenvector. Therefore crucial to the argument is the following consequence of the Hermiticity of A: If A is Hermitian and e is an eigenvector of A, then not only is the linear span of e an invariant subspace of A, but so is its orthogonal complement.
The argument is also valid for symmetric operators on finite-dimensional real inner product spaces, but the existence of an eigenvector is harder to establish. A real symmetric matrix has real eigenvalues, therefore eigenvectors with real entries.
The spectral decomposition of an operator A which has an orthonormal basis of eigenvectors is obtained by grouping together all vectors corresponding to the same eigenvalue. Thus
Note that these spaces are invariantly defined, in that the definition does not depend on any choice of specific eigenvectors.
As an immediate consequence of the spectral theorem for symmetric operators we get the spectral decomposition theorem: V is the orthogonal direct sum of the spaces Vλ where the index ranges over eigenvalues. Another equivalent formulation, letting Pλ be the orthogonal projection onto Vλ (
) and λ1,..., λm the eigenvalues of A, is
The spectral decomposition is a special case of the Schur decomposition. It is also a special case of the singular value decomposition.
If A is a real symmetric matrix, it follows by the real version of the spectral theorem for symmetric operators that there is an orthogonal matrix U such that UAUT is diagonal and all the eigenvalues of A are real.
In other words, A is normal if and only if there exists a unitary matrix U such that
where Λ is the diagonal matrix the entries of which 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 Λ need not be real.
Theorem. Suppose A is a compact self-adjoint operator on a Hilbert space V. 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. To prove this, we cannot rely on determinants to show existence of eigenvalues, but instead one can use a maximization argument analogous to the variational characterization of eigenvalues. The above spectral theorem holds for real or complex Hilbert spaces.
If the compactness assumption is removed, it is not true that a self adjoint operator has eigenvectors.
The next generalization we consider is that of bounded self-adjoint operators A on a Hilbert space V. Such operators may have no eigenvalues: for instance let A be the operator multiplication by t on L2[0, 1], that is
Theorem. Let A be a bounded self-adjoint operator on a Hilbert space H. Then there is a measure space (X, Σ, μ) and a real-valued measurable function f on X and a unitary operator U:H → L2μ(X) such that
where T is the multiplication operator:
This is the beginning of the vast research area of functional analysis called operator theory.
There is also an analogous spectral theorem for normal operators on Hilbert spaces. In this case it is more common to express the spectral theorem as an integral of the coordinate function over the spectrum with respect to a projection-valued measure.
When the normal operator in question is compact, this spectral theorem reduces to the finite-dimensional spectral theorem above, except that the operator is expressed as a linear combination of possibly infinitely many projections.
Canonical is an adjective derived from . Canon comes from the Greek word kanon "rule" (perhaps originally from kanna "reed", cognate to cane) is used in various meanings.
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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, or eigendecomposition, of the underlying vector space on which it acts.
In this article we consider mainly 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 matrices
We begin by considering a Hermitian matrix A on a finite-dimensional real or complex inner product space V with the standard Hermitian inner product; in Dirac's bra-ket notation, the Hermitian condition means
for all x, y elements of V.
An equivalent condition is that A* = A, where A* is the conjugate transpose of A. If A is a real matrix, this is also equivalent to AT = A
Recall that an eigenvector of a linear operator A is a (non-zero) vector x such that Ax = λx for some scalar λ. The value λ is the corresponding eigenvalue.
Theorem. There is an orthonormal basis of V consisting of eigenvectors of A. Each eigenvalue is real.
This result is of such importance in many parts of mathematics, that we provide a sketch of a proof for the case wherein the underlying field of scalars is the complex numbers. First we show that all the eigenvalues are real. Suppose that λ is an eigenvalue of A with corresponding eigenvector x. Thus
Since x is non-zero, it follows that λ equals its own conjugate and is therefore real.
To prove the existence of an eigenvector basis, we use induction on the dimension of V. In fact it suffices to show A has at least one non-zero eigenvector e. For then we can consider the space K of vectors v orthogonal to e. This is finite-dimensional because it is a subspace of a finite dimensional space, and A has the property that it maps every vector w in K into K. This is shown as follows: If w ∈ K, then using the symmetry property of A,
Moreover, A considered as a linear operator on K is also symmetric, so by the induction hypothesis there is a basis for V consisting of eigenvectors of A.
It remains, however, to show that A has at least one eigenvector. Since the ground field is algebraically closed, the polynomial function (called the characteristic polynomial of A)
has a complex root r. This implies the linear operator A − rI is not invertible and hence maps a non-zero vector e to 0. This vector e is a non-zero eigenvector of A. This implies that r is an eigenvalue, so is actually a real number. This completes the proof.
Notice the second part of the proof works for any square matrices. Clearly any square matrix has at least one eigenvector. Therefore crucial to the argument is the following consequence of the Hermiticity of A: If A is Hermitian and e is an eigenvector of A, then not only is the linear span of e an invariant subspace of A, but so is its orthogonal complement.
The argument is also valid for symmetric operators on finite-dimensional real inner product spaces, but the existence of an eigenvector is harder to establish. A real symmetric matrix has real eigenvalues, therefore eigenvectors with real entries.
The spectral decomposition of an operator A which has an orthonormal basis of eigenvectors is obtained by grouping together all vectors corresponding to the same eigenvalue. Thus
Note that these spaces are invariantly defined, in that the definition does not depend on any choice of specific eigenvectors.
As an immediate consequence of the spectral theorem for symmetric operators we get the spectral decomposition theorem: V is the orthogonal direct sum of the spaces Vλ where the index ranges over eigenvalues. Another equivalent formulation, letting Pλ be the orthogonal projection onto Vλ (
) and λ1,..., λm the eigenvalues of A, is
The spectral decomposition is a special case of the Schur decomposition. It is also a special case of the singular value decomposition.
If A is a real symmetric matrix, it follows by the real version of the spectral theorem for symmetric operators that there is an orthogonal matrix U such that UAUT is diagonal and all the eigenvalues of A are real.
Normal matrices
The spectral theorem extends to a more general class of matrices. Let A be an operator on a finite-dimensional inner product space. A is said to be normal if A* A = A A*. One can show that A is normal if and only if it is unitarily diagonalizable: By the Schur decomposition, we have A = U T U*, where U is unitary and T upper-triangular. Since A is normal, T T* = T* T. Therefore T must be diagonal. The converse is also obvious.In other words, A is normal if and only if there exists a unitary matrix U such that
where Λ is the diagonal matrix the entries of which 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 Λ need not be real.
The spectral theorem for compact self-adjoint operators
Theorem. Suppose A is a compact self-adjoint operator on a Hilbert space V. 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. To prove this, we cannot rely on determinants to show existence of eigenvalues, but instead one can use a maximization argument analogous to the variational characterization of eigenvalues. The above spectral theorem holds for real or complex Hilbert spaces.
If the compactness assumption is removed, it is not true that a self adjoint operator has eigenvectors.
Functional analysis
- See also: eigenfunction
The next generalization we consider is that of bounded self-adjoint operators A on a Hilbert space V. Such operators may have no eigenvalues: for instance let A be the operator multiplication by t on L2[0, 1], that is
Theorem. Let A be a bounded self-adjoint operator on a Hilbert space H. Then there is a measure space (X, Σ, μ) and a real-valued measurable function f on X and a unitary operator U:H → L2μ(X) such that
where T is the multiplication operator:
This is the beginning of the vast research area of functional analysis called operator theory.
There is also an analogous spectral theorem for normal operators on Hilbert spaces. In this case it is more common to express the spectral theorem as an integral of the coordinate function over the spectrum with respect to a projection-valued measure.
When the normal operator in question is compact, this spectral theorem reduces to the finite-dimensional spectral theorem above, except that the operator is expressed as a linear combination of possibly infinitely many projections.
The spectral theorem for general self-adjoint operators
Many important linear operators which occur in analysis, such as differential operators, are unbounded. There is however a spectral theorem for self-adjoint operators that applies in many of these cases. To give an example, any constant coefficient differential operator is unitarily equivalent to a multiplication operator. Indeed the unitary operator that implements this equivalence is the Fourier transform.See also
- Matrix decomposition
- Jordan decomposition, of which the spectral decomposition is a special case.
- Singular value decomposition, a generalisation of spectral theorem to arbitrary matrices.
- Eigendecomposition (matrix)
References
- Sheldon Axler, Linear Algebra Done Right, Springer Verlag, 1997
- Paul Halmos, "What Does the Spectral Theorem Say?", American Mathematical Monthly, volume 70, number 3 (1963), pages 241-247
Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations.
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Functional analysis is the branch of mathematics, and specifically of analysis, concerned with the study of vector spaces and operators acting upon them. It has its historical roots in the study of functional spaces, in particular transformations of functions, such as the Fourier
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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theorem is a statement, often stated in natural language, that can be proved on the basis of explicitly stated or previously agreed assumptions. In logic, a theorem is a statement in a formal language that can be derived by applying rules and axioms from a deductive system.
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operator is a function, that operates on (or modifies) another function. Often, an "operator" is a function that acts on functions to produce other functions (the sense in which Oliver Heaviside used the term); or it may be a generalization of such a function, as in linear algebra,
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In linear algebra, a diagonal matrix is a square matrix in which the entries outside the main diagonal are all zero. The diagonal entries themselves may or may not be zero.
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In mathematics, a linear map (also called a linear transformation or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication.
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In operator theory, a multiplication operator is a linear operator T defined on some vector space of functions and whose value at a function φ is given by multiplication by a fixed function f.
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In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix. The name was introduced by David Hilbert in his original formulation of Hilbert space theory, which was cast in terms of quadratic forms in
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In mathematics, on a finite-dimensional inner product space, a self-adjoint operator is one that is its own adjoint, or, equivalently, one whose matrix is Hermitian, where a Hermitian matrix is one which is equal to its own conjugate transpose.
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In mathematics, especially functional analysis, a normal operator on a Hilbert space (or more generally in a C* algebra) is a continuous linear operator
that commutes with its hermitian adjoint ''N*:
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that commutes with its hermitian adjoint ''N*:
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Hilbert space, named after the David Hilbert, generalizes the notion of Euclidean space in a way that extends methods of vector algebra from the two-dimensional plane and three-dimensional space to infinite-dimensional spaces.
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For the software company, see .
Canonical is an adjective derived from . Canon comes from the Greek word kanon "rule" (perhaps originally from kanna "reed", cognate to cane) is used in various meanings.
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In mathematics, an element x of a star-algebra is self-adjoint if .
A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation.
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A collection C of elements of a star-algebra is self-adjoint if it is closed under the involution operation.
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A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the j
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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inner product space is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry.
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inner product space is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry.
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Bra-ket notation is the standard notation for describing quantum states in the theory of quantum mechanics. It can also be used to denote abstract vectors and linear functionals in pure mathematics.
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conjugate transpose, Hermitian transpose, or adjoint matrix of an m-by-n matrix A with complex entries is the n-by-m matrix A* obtained from A
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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eigenvector of the transformation and the blue vector is not. Since the red vector was neither stretched nor compressed, its eigenvalue is 1. All vectors with the same vertical direction - i.e., parallel to this vector - are also eigenvectors, with the same eigenvalue.
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Definition
In mathematics, an orthonormal basis of an inner product space V (i.e., a vector space with an inner product), or in particular of a Hilbert space H..... Click the link for more information.
In mathematics, a complex number is a number of the form
where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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where a and b are real numbers, and i is the imaginary unit, with the property i ² = −1.
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In mathematics, a field is said to be algebraically closed if every polynomial in one variable of degree at least , with coefficients in , has a zero (root) in .
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Examples
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characteristic polynomial. This polynomial encodes several important properties of the matrix, most notably its eigenvalues, its determinant and its trace.
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Motivation
Given a square matrix , we want to find a polynomial whose roots are precisely the eigenvalues of ...... Click the link for more information.
projection is a linear transformation P from a vector space to itself such that P2 = P. Projections map the whole vector space to a subspace and leave the points in that subspace unchanged.
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In the mathematical discipline of linear algebra, the Schur decomposition or Schur triangulation (named after Issai Schur) is an important matrix decomposition.
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Statement
The Schur decomposition reads as follows: if A is a n × n..... Click the link for more information.
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