Spectral theorem - Citizendia

In mathematics, particularly linear algebra and functional analysis, the spectral theorem is any of a number of results about linear operators or about matrices. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Linear algebra is the branch of Mathematics concerned with For functional analysis as used in psychology see the Functional analysis (psychology article In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally 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). In Mathematics, a theorem is a statement proven on the basis of previously accepted or established statements In Mathematics, an operator is a function which operates on (or modifies another function In Linear algebra, a Square matrix A is called diagonalizable if it is similar to a Diagonal matrix, i In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero 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. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Operator theory, a multiplication operator is a Linear operator T defined on some vector space of functions and whose value at a function In more abstract language, the spectral theorem is a statement about commutative C*-algebras. C*-algebras (pronounced "C-star" are an important area of research in Functional analysis, a branch of Mathematics. See also spectral theory for a historical perspective. In Mathematics, spectral theory is an inclusive term for theories extending the Eigenvector and Eigenvalue theory of a single Square matrix.

Examples of operators to which the spectral theorem applies are self-adjoint operators or more generally normal operators on Hilbert spaces. 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 In Mathematics, especially Functional analysis, a normal operator on a Hilbert space H (or more generally in a C* algebra) is a This article assumes some familiarity with Analytic geometry and the concept of a limit.

The spectral theorem also provides a canonical decomposition, called the spectral decomposition, eigenvalue decomposition, or eigendecomposition, of the underlying vector space on which it acts. Canonical is an Adjective derived from canon. Canon comes from the Greek word kanon, "rule" (perhaps originally from In the mathematical discipline of Linear algebra, eigendecomposition or sometimes Spectral decomposition is the Factorization of

In this article we consider mainly the simplest kind of spectral theorem, that for a self-adjoint operator on a Hilbert space. In Mathematics, an element x of a Star-algebra is self-adjoint if x^*=x However, as noted above, the spectral theorem also holds for normal operators on a Hilbert space.

1 Finite-dimensional case
- 1.1 Hermitian matrices
- 1.2 Normal matrices
2 The spectral theorem for compact self-adjoint operators
3 The spectral theorem for bounded self-adjoint operators
4 The spectral theorem for general self-adjoint operators
5 See also
6 References

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; the Hermitian condition means

$\langle A x ,\, y \rangle = \langle x ,\, A y \rangle$

for all x, y elements of V. A Hermitian matrix (or self-adjoint matrix) is a Square matrix with complex entries which is equal to its own Conjugate transpose &mdash that In Mathematics, the real numbers may be described informally in several different ways Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product.

An equivalent condition is that A* = A, where A* is the conjugate transpose of A. In Mathematics, the conjugate transpose, Hermitian transpose, or adjoint matrix of an m -by- n matrix A with If A is a real matrix, this is equivalent to A^T = A (that is, A is a symmetric matrix). In Linear algebra, a symmetric matrix is a Square matrix, A, that is equal to its Transpose A = A^{T} The eigenvalues of a Hermitian matrix are real.

Recall that an eigenvector of a linear operator A is a (non-zero) vector x such that Ax = λx for some scalar λ. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes The value λ is the corresponding eigenvalue. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes

Theorem. There is an orthonormal basis of V consisting of eigenvectors of A. In Mathematics, an orthonormal basis of an Inner product space V (i Each eigenvalue is real.

We provide a sketch of a proof for the case where the underlying field of scalars is the complex numbers. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted

By the fundamental theorem of algebra, any square matrix with complex entries has at least one eigenvector. In Mathematics, the Fundamental theorem of algebra states that every non-constant single-variable Polynomial with complex coefficients has at Now if A is Hermitian with eigenvector e₁, we can consider the space K = span{e₁}^⊥, the orthogonal complement of e₁ . By Hermiticity, K is an invariant subspace of A. In Mathematics, an invariant subspace of a Linear mapping T: V &rarr V from some Vector space Applying the same argument to K shows that A has an eigenvector e₂ ∈ K. Finite induction then finishes the proof.

The spectral theorem holds also for symmetric matrices 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.

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

$V_\lambda = \{\,v \in V: A v = \lambda v\,\}$

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. Let P_λ be the orthogonal projection onto V_λ and λ₁,. . . , λ_m the eigenvalues of A, one can write its spectral decomposition thus:

$A =\lambda_1 P_{\lambda_1} +\cdots+\lambda_m P_{\lambda_m}.$

The spectral decomposition is a special case of the Schur decomposition. In the mathematical discipline of Linear algebra, the Schur decomposition or Schur triangulation (named after Issai Schur) is an important It is also a special case of the singular value decomposition. In Linear algebra, the singular value decomposition ( SVD) is an important factorization of a rectangular real or complex matrix

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^*. In Mathematics, especially Functional analysis, a normal operator on a Hilbert space H (or more generally in a C* algebra) is 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. In the mathematical discipline of Linear algebra, the Schur decomposition or Schur triangulation (named after Issai Schur) is an important 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

$A=U \Lambda U^* \;$

where Λ is the diagonal matrix the entries of which are the eigenvalues of A. In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* In Linear algebra, a diagonal matrix is a Square matrix in which the entries outside the Main diagonal (↘ are all zero In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes 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

Main article: Compact operator on Hilbert space

In Hilbert spaces in general, the statement of the spectral theorem for compact self-adjoint operators is virtually the same as in the finite-dimensional case. In Functional analysis, Compact operators on Hilbert spaces are a direct extension of matrices in the Hilbert spaces they are precisely the closure of Finite In Functional analysis, a branch of Mathematics, a compact operator is a Linear operator L from a Banach space X to another 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

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. In Mathematics, an orthonormal basis of an Inner product space V (i 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 every self adjoint operator has eigenvectors.

The spectral theorem for bounded self-adjoint operators

See also: eigenfunction

The next generalization we consider is that of bounded self-adjoint operators A on a Hilbert space V. In Mathematics, an eigenfunction of a Linear operator, A, defined on some Function space is any non-zero function f in In Functional analysis (a branch of Mathematics) a bounded linear operator is a Linear transformation L between Normed vector spaces Such operators may have no eigenvalues: for instance let A be the operator multiplication by t on L²[0, 1], that is

$[A \varphi](t) = t \varphi(t). \;$

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 → L²_μ(X) such that

$U^* T U = A \;$

where T is the multiplication operator:

$[T \varphi](x) = f(x) \varphi(x). \;$

This is the beginning of the vast research area of functional analysis called operator theory. In Mathematics the concept of a measure generalizes notions such as "length" "area" and "volume" (but not all of its applications have to do with In Operator theory, a multiplication operator is a Linear operator T defined on some vector space of functions and whose value at a function In Mathematics, operator theory is the branch of Functional analysis which deals with Bounded linear operators and their properties

There is also an analogous spectral theorem for bounded normal operators on Hilbert spaces. In Mathematics, especially Functional analysis, a normal operator on a Hilbert space H (or more generally in a C* algebra) is a The only difference in the conclusion is that now $f$ may be complex-valued.

An alternative 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 projection-valued measure. In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes In Mathematics, particularly Functional analysis a projection-valued measure is a function defined on certain subsets of a fixed set and whose values are self-adjoint When the normal operator in question is compact, this version of the 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. In Functional analysis, a branch of Mathematics, a compact operator is a Linear operator L from a Banach space X to another

The spectral theorem for general self-adjoint operators

Many important linear operators which occur in analysis, such as differential operators, are unbounded. Analysis has its beginnings in the rigorous formulation of Calculus. In Mathematics, a differential operator is an Operator defined as a function of the differentiation operator There is however a spectral theorem for self-adjoint operators that applies in many of these cases. 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 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; the multiplication operator is a type of Fourier multiplier. This article specifically discusses Fourier transformation of functions on the Real line; for other kinds of Fourier transformation see Fourier analysis and In Fourier analysis, a Fourier multiplier (or multiplier for short is a kind of Linear operator, or transformation of functions.

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

In the mathematical discipline of Linear algebra, eigendecomposition or sometimes Spectral decomposition is the Factorization of Paul Richard Halmos ( March 3 1916 &mdash October 2 2006) was a Hungarian -born Jewish American Mathematician

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