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{{Short description|Result about when a matrix can be diagonalized}}
In [[mathematics]], particularly [[linear algebra]] and [[functional analysis]], a '''spectral theorem''' is a result about when a [[linear operator]] or [[matrix (mathematics)|matrix]] can be [[Diagonalizable matrix|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 finite-dimensional vector spaces but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of [[linear operator]]s that can be modeled by [[multiplication operator]]s, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative [[C*-algebra]]s. See also [[spectral theory]] for a historical perspective.
In [[linear algebra]] and [[functional analysis]], a '''spectral theorem''' is a result about when a [[linear operator]] or [[matrix (mathematics)|matrix]] can be [[Diagonalizable matrix|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 [[finite-dimensional vector space]]s but requires some modification for operators on infinite-dimensional spaces. In general, the spectral theorem identifies a class of [[linear operator]]s that can be modeled by [[multiplication operator]]s, which are as simple as one can hope to find. In more abstract language, the spectral theorem is a statement about commutative [[C*-algebra]]s. See also [[spectral theory]] for a historical perspective.


Examples of operators to which the spectral theorem applies are [[self-adjoint operator]]s or more generally [[normal operator]]s on [[Hilbert space]]s.
Examples of operators to which the spectral theorem applies are [[self-adjoint operator]]s or more generally [[normal operator]]s on [[Hilbert space]]s.


The spectral theorem also provides a [[canonical form|canonical]] decomposition, called the '''spectral decomposition''', '''eigenvalue decomposition''', or '''[[eigendecomposition of a matrix|eigendecomposition]]''', of the underlying vector space on which the operator acts.
The spectral theorem also provides a [[canonical form|canonical]] decomposition, called the '''[[eigendecomposition of a matrix|spectral decomposition]]''', of the underlying vector space on which the operator acts.


[[Augustin-Louis Cauchy]] proved the spectral theorem for [[Hermitian matrix|self-adjoint matrices]], i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about determinants.<ref>{{cite journal|url = http://www.sciencedirect.com/science/article/pii/0315086075900324 | doi=10.1016/0315-0860(75)90032-4 | volume=2 | title=Cauchy and the spectral theory of matrices | year=1975 | journal=Historia Mathematica | pages=1–29 | last1 = Hawkins | first1 = Thomas}}</ref><ref>[http://www.mathphysics.com/opthy/OpHistory.html A Short History of Operator Theory by Evans M. Harrell II]</ref> The spectral theorem as generalized by [[John von Neumann]] is today perhaps the most important result of operator theory.
[[Augustin-Louis Cauchy]] proved the spectral theorem for [[Symmetric matrix|symmetric matrices]], i.e., that every real, symmetric matrix is diagonalizable. In addition, Cauchy was the first to be systematic about [[determinant]]s.<ref>{{cite journal| doi=10.1016/0315-0860(75)90032-4 | volume=2 | title=Cauchy and the spectral theory of matrices | year=1975 | journal=Historia Mathematica | pages=1–29 | last1 = Hawkins | first1 = Thomas| doi-access=free }}</ref><ref>[http://www.mathphysics.com/opthy/OpHistory.html A Short History of Operator Theory by Evans M. Harrell II]</ref> 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.
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 ==<!-- This section is linked from [[Singular value decomposition]] -->
== Finite-dimensional case ==
<!-- This section is linked from [[Singular value decomposition]] -->


=== Hermitian maps and Hermitian matrices ===
=== Hermitian maps and Hermitian matrices ===
We begin by considering a [[Hermitian matrix]] on <math>\mathbb{C}^n</math> (but the following discussion will be adaptable to the more restrictive case of [[symmetric matrix|symmetric matrices]] on <math>\mathbb{R}^n</math>). We consider a [[Hermitian operator|Hermitian map]] {{math|''A''}} on a finite-dimensional [[complex number|complex]] [[inner product space]] {{math|''V''}} endowed with a [[Definite bilinear form|positive definite]] [[sesquilinear form|sesquilinear]] [[inner product]] <math>\langle\cdot,\cdot\rangle</math>. The Hermitian condition on <math>A</math> means that for all {{math|''x'', ''y'' ∈ ''V''}},
We begin by considering a [[Hermitian matrix]] on <math>\mathbb{C}^n</math> (but the following discussion will be adaptable to the more restrictive case of [[symmetric matrix|symmetric matrices]] on {{nobr|<math>\mathbb{R}^n</math>).}} We consider a [[Hermitian operator|Hermitian map]] {{math|''A''}} on a finite-dimensional [[complex number|complex]] [[inner product space]] {{math|''V''}} endowed with a [[Definite bilinear form|positive definite]] [[sesquilinear form|sesquilinear]] [[inner product]] <math>\ \langle \cdot, \cdot \rangle ~.</math> The Hermitian condition on <math>\ A\ </math> means that for all {{math|''x'', ''y'' ∈ ''V''}},
<math display="block">\ \langle\ A x, y\ \rangle = \langle\ x, A y\ \rangle ~.</math>


An equivalent condition is that {{math| ''A''{{sup|*}} {{=}} ''A'' }}, where {{math| ''A''{{sup|*}} }} is the [[Hermitian conjugate]] of {{math|''A''}}. In the case that {{math|''A''}} is identified with a Hermitian matrix, the matrix of {{math| ''A''{{sup|*}} }} is equal to its [[conjugate transpose]]. (If {{math|''A''}} is a [[real matrix]], then this is equivalent to {{math| ''A''{{sup|T}} {{=}} ''A''}}, that is, {{math|''A''}} is a [[symmetric matrix]].)
:<math> \langle A x ,\, y \rangle = \langle x ,\, A y \rangle .</math>


(An equivalent condition is that {{math|1=''A''<sup>∗</sup> = ''A''}}, where {{math|''A''<sup>∗</sup>}} is the [[hermitian conjugate]] of {{math|''A''}}.) In the case that {{math|''A''}} is identified with a Hermitian matrix, the matrix of {{math|''A''<sup>∗</sup>}} can be identified with its [[conjugate transpose]]. (If {{math|''A''}} is a [[real matrix]], this is equivalent to {{math|1=''A''<sup>T</sup> = ''A''}}, that is, {{math|''A''}} is a [[symmetric matrix]].)
This condition implies that all eigenvalues of a Hermitian map are real: To see this, it is enough to apply it to the case when {{math| ''x'' {{=}} ''y''}} is an eigenvector. (Recall that an [[eigenvector]] of a linear map {{math|''A''}} is a non-zero vector {{math|''v''}} such that {{math| ''A v'' {{=}} ''λv''}} for some scalar {{math|''λ''}}. The value {{math|''λ''}} is the corresponding [[eigenvalue]]. Moreover, the [[eigenvalues]] are roots of the [[characteristic polynomial]].)


{{math theorem | math_statement = If {{math|''A''}} is Hermitian on {{math|''V''}}, then there exists an [[orthonormal basis]] of {{math|''V''}} consisting of eigenvectors of {{math|''A''}}. Each eigenvalue of {{math|''A''}} is real.}}
This condition implies that all eigenvalues of a Hermitian map are real: it is enough to apply it to the case when {{math|1=''x'' = ''y''}} is an eigenvector. (Recall that an [[eigenvector]] of a linear map {{math|''A''}} is a (non-zero) vector {{math|''x''}} such that {{math|1=''Ax'' = ''λx''}} for some scalar {{math|''λ''}}. The value {{math|''λ''}} is the corresponding [[eigenvalue]]. Moreover, the [[eigenvalues]] are solutions to the [[characteristic polynomial]].)

'''Theorem'''. If {{math|''A''}} is Hermitian, there exists an [[orthonormal basis]] of {{math|''V''}} consisting of eigenvectors of {{math|''A''}}. Each eigenvalue is real.


We provide a sketch of a proof for the case where the underlying field of scalars is the [[complex number]]s.
We provide a sketch of a proof for the case where the underlying field of scalars is the [[complex number]]s.


By the [[fundamental theorem of algebra]], applied to the [[characteristic polynomial]] of {{math|''A''}}, there is at least one eigenvalue {{math|''λ''<sub>1</sub>}} and eigenvector {{math|''e''<sub>1</sub>}}. Then since
By the [[fundamental theorem of algebra]], applied to the [[characteristic polynomial]] of {{math|''A''}}, there is at least one complex eigenvalue {{math| ''λ''{{sub|1}} }} and corresponding eigenvector {{nobr|{{math| ''v''{{sub|1}} }} ,}} which must by definition be non-zero. Then since
:<math>\lambda_1 \langle e_1, e_1 \rangle = \langle A (e_1), e_1 \rangle = \langle e_1, A(e_1) \rangle = \bar\lambda_1 \langle e_1, e_1 \rangle </math>
<math display="block">\ \lambda_1\ \langle\ v_1, v_1\ \rangle = \langle\ A (v_1), v_1\ \rangle = \langle\ v_1, A(v_1)\ \rangle = \bar\lambda_1\ \langle\ v_1, v_1\ \rangle\ ,</math>
we find that {{math|''λ''<sub>1</sub>}} is real. Now consider the space {{math|1=''K'' = span{''e''<sub>1</sub>}<sup>⊥</sup>}}, the [[orthogonal complement]] of {{math|''e''<sub>1</sub>}}. By Hermiticity, {{math|''K''}} is an [[invariant subspace]] of {{math|''A''}}. Applying the same argument to {{math|''K''}} shows that {{math|''A''}} has an eigenvector {{math|''e''<sub>2</sub> ''K''}}. Finite induction then finishes the proof.
we find that {{math| ''λ''{{sub|1}} }} is real. Now consider the space <math>\ \mathcal{K}^{n-1} = \text{span}\left(\ v_1\ \right)^\perp\ ,</math> the [[orthogonal complement]] of {{nobr|{{math| ''v''<sub>1</sub>}} .}} By Hermiticity, <math>\ \mathcal{K}^{n-1}\ </math> is an [[invariant subspace]] of {{math|''A''}}. To see that, consider any <math>\ k \in \mathcal{K}^{n-1}</math> so that <math>\ \langle\ k, v_1\ \rangle = 0\ </math> by definition of <math>\mathcal{K}^{n-1} ~.</math> To satisfy invariance, we need to check if <math>\ A(k) \in \mathcal{K}^{n-1} ~.</math> This is true because <math>\ \langle\ A(k), v_1\ \rangle = \langle\ k, A(v_1)\ \rangle = \langle\ k, \lambda_1\ v_1\ \rangle = 0 ~.</math> Applying the same argument to <math>\ \mathcal{K}^{n-1}\ </math> shows that {{math|''A''}} has at least one real eigenvalue <math>\lambda_2</math> and corresponding eigenvector <math>\ v_2 \in \mathcal{K}^{n-1} \perp v_1 ~.</math> This can be used to build another invariant subspace <math>\ \mathcal{K}^{n-2} = \text{span}\left(\ \{v_1, v_2\}\ \right)^\perp ~.</math> 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]]. To prove this, consider {{math|''A''}} as a Hermitian matrix and use the fact that all eigenvalues of a Hermitian matrix are real.

If one chooses the eigenvectors of {{math|''A''}} as an orthonormal basis, the matrix representation of {{math|''A''}} in this basis is diagonal. Equivalently, {{math|''A''}} can be written as a linear combination of pairwise orthogonal projections, called its '''spectral decomposition'''. Let

:<math> V_\lambda = \{\,v \in V: A v = \lambda v\,\}</math>


The matrix representation of {{math|''A''}} 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. {{math|''A''}} can be written as a linear combination of pairwise orthogonal projections, called its '''spectral decomposition'''. Let
be the eigenspace corresponding to an eigenvalue {{math|''λ''}}. Note that the definition does not depend on any choice of specific eigenvectors. {{math|''V''}} is the orthogonal direct sum of the spaces {{math|''V''<sub>''λ''</sub>}} where the index ranges over eigenvalues. Let {{math|''P''<sub>''λ''</sub>}} be the [[Orthogonal projection#Orthogonal projections|orthogonal projection]] onto {{math|''V''<sub>''λ''</sub>}} and {{math|''λ''<sub>1</sub>, ..., ''λ''<sub>''m''</sub>}} the eigenvalues of {{math|''A''}}, one can write its spectral decomposition thus:
<math display="block">\ V_{\lambda} = \{\ v \in V\ :\ A\ v = \lambda\ v\ \}\ </math>
be the eigenspace corresponding to an eigenvalue <math>\ \lambda ~.</math> Note that the definition does not depend on any choice of specific eigenvectors. In general, {{math|''V''}} is the orthogonal direct sum of the spaces <math>\ V_{\lambda}\ </math> where the <math>\ \lambda\ </math> ranges over the [[Spectrum of a matrix|spectrum]] of <math>\ A ~.</math>


When the matrix being decomposed is Hermitian, the spectral decomposition is a special case of the [[Schur decomposition]] (see the proof in case of [[#Normal matrices|normal matrices]] below).
:<math>A =\lambda_1 P_{\lambda_1} +\cdots+\lambda_m P_{\lambda_m}.</math>


The spectral decomposition is a special case of both the [[Schur decomposition]] and the [[singular value 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 <math>\ A \in \mathbb{C}^{m \times n}\ </math> can be expressed as
<math>\ A = U\ \Sigma\ V^{*}\ ,</math> where <math>\ U \in \mathbb{C}^{m \times m}\ </math> and <math>\ V \in \mathbb{C}^{n \times n}\ </math> are [[unitary matrices]] and <math>\ \Sigma \in \mathbb{R}^{m \times n}\ </math> is a diagonal matrix. The diagonal entries of <math>\ \Sigma\ </math> are uniquely determined by <math>\ A\ </math> and are known as the [[singular values]] of <math>\ A ~.</math> If <math>\ A\ </math> is Hermitian, then <math>\ A^* = A\ </math> and <math>\ V\ \Sigma\ U^* = U\ \Sigma\ V^*\ </math> which implies <math>\ U = V ~.</math>
=== Normal matrices ===
=== Normal matrices ===
{{main|Normal matrix}}
{{main|Normal matrix}}
The spectral theorem extends to a more general class of matrices. Let {{math|''A''}} be an operator on a finite-dimensional inner product space. {{math|''A''}} is said to be [[normal matrix|normal]] if {{math|1=''A''<sup>∗</sup>''A'' = ''AA''<sup>∗</sup>}}. One can show that {{math|''A''}} is normal if and only if it is unitarily diagonalizable. Proof: By the [[Schur decomposition]], we can write any matrix as {{math|1=''A'' = ''UTU''<sup>∗</sup>}}, where {{math|''U''}} is unitary and {{math|''T''}} is upper-triangular.
The spectral theorem extends to a more general class of matrices. Let {{math|''A''}} be an operator on a finite-dimensional inner product space. {{math|''A''}} is said to be [[normal matrix|normal]] if {{nobr|{{math| ''A''{{sup|*}} ''A'' {{=}} ''A A''{{sup|*}} }} .}}
One can show that {{math|''A''}} is normal if and only if it is unitarily diagonalizable using the [[Schur decomposition]]. That is, any matrix can be written as {{nobr|{{math| ''A'' {{=}} ''U T U''{{sup|*}} }} ,}} where {{math| ''U'' }} is unitary and {{math| ''T'' }} is [[upper triangular]].
If {{math|''A''}} is normal, one sees that {{math|1=''TT''<sup></sup> = ''T''<sup>*</sup>''T''}}. Therefore, {{math|''T''}} must be diagonal since a normal upper triangular matrix is diagonal (see [[normal matrix#Consequences|normal matrix]]). The converse is obvious.
If {{math|''A''}} is normal, then one sees that {{nobr|{{math| ''T T''<sup>*</sup> {{=}} ''T''{{sup|*}} ''T''}} .}} Therefore, {{math|''T''}} must be diagonal since a normal upper triangular matrix is diagonal (see [[normal matrix#Consequences|normal matrix]]). The converse is obvious.


In other words, {{math|''A''}} is normal if and only if there exists a [[unitary matrix]] {{math|''U''}} such that
In other words, {{math|''A''}} is normal if and only if there exists a [[unitary matrix]] {{math|''U''}} such that
<math display="block">\ A = U\ D\ U^*\ ,</math>

:<math>A=U D U^*,</math>

where {{math|''D''}} is a [[diagonal matrix]]. Then, the entries of the diagonal of {{math|''D''}} are the [[eigenvalue]]s of {{math|''A''}}. The column vectors of {{math|''U''}} are the eigenvectors of {{math|''A''}} and they are orthonormal. Unlike the Hermitian case, the entries of {{math|''D''}} need not be real.
where {{math|''D''}} is a [[diagonal matrix]]. Then, the entries of the diagonal of {{math|''D''}} are the [[eigenvalue]]s of {{math|''A''}}. The column vectors of {{math|''U''}} are the eigenvectors of {{math|''A''}} and they are orthonormal. Unlike the Hermitian case, the entries of {{math|''D''}} need not be real.


== Compact self-adjoint operators ==
== Compact self-adjoint operators ==
{{main|Compact operator on Hilbert space}}
{{see also|Compact operator on Hilbert space#Spectral theorem}}
In the more general setting of Hilbert spaces, which may have an infinite dimension, the statement of the spectral theorem for [[compact operator|compact]] [[self-adjoint operators]] is virtually the same as in the finite-dimensional case.
In the more general setting of Hilbert spaces, which may have an infinite dimension, the statement of the spectral theorem for [[compact operator|compact]] [[self-adjoint operators]] is virtually the same as in the finite-dimensional case.


'''Theorem'''. Suppose {{math|''A''}} is a compact self-adjoint operator on a (real or complex) Hilbert space {{math|''V''}}. Then there is an [[orthonormal basis]] of {{math|''V''}} consisting of eigenvectors of {{math|''A''}}. Each eigenvalue is real.
{{math theorem | math_statement =Suppose {{math|''A''}} is a compact self-adjoint operator on a (real or complex) Hilbert space {{math|''V''}}. Then there is an [[orthonormal basis]] of {{math|''V''}} consisting of eigenvectors of {{math|''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.
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 self-adjoint operator has eigenvectors.
If the compactness assumption is removed, then it is ''not'' true that every self-adjoint operator has eigenvectors. For example, the multiplication operator <math>M_{x}</math> on <math>L^2([0,1])</math> which takes each <math>\psi(x) \in L^2([0,1])</math> to <math>x\psi(x)</math> is bounded and self-adjoint, but has no eigenvectors. However, its spectrum, suitably defined, is still equal to <math>[0,1]</math>, see [[Spectrum_(functional_analysis)#Spectrum_of_a_bounded_operator| spectrum of bounded operator]].


== Bounded self-adjoint operators ==
== Bounded self-adjoint operators ==
Line 66: Line 65:
===Possible absence of eigenvectors===
===Possible absence of eigenvectors===


The next generalization we consider is that of [[bounded operator|bounded]] self-adjoint operators on a Hilbert space. Such operators may have no eigenvalues: for instance let {{math|''A''}} be the operator of multiplication by {{math|''t''}} on {{math|''L''<sup>2</sup>[0, 1]}}, that is,<ref>{{harvnb|Hall|2013}} Section 6.1</ref>
The next generalization we consider is that of [[Self-adjoint_operator#Bounded_self-adjoint_operators|bounded self-adjoint operators]] on a Hilbert space. Such operators may have no eigenvectors: for instance let {{math|''A''}} be the operator of multiplication by {{math|''t''}} on <math>L^2([0,1])</math>, that is,<ref>{{harvnb|Hall|2013}} Section 6.1</ref>
<math display="block"> [A f](t) = t f(t). </math>


This operator does not have any eigenvectors ''in'' <math>L^2([0,1])</math>, though it does have eigenvectors in a larger space. Namely the [[Distribution (mathematics)|distribution]] <math>f(t)=\delta(t-t_0)</math>, where <math>\delta</math> 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 {{math|''L''<sup>2</sup>[0, 1]}} or any other [[Banach space]]. Thus, the delta-functions are "generalized eigenvectors" of <math>A</math> but not eigenvectors in the usual sense.
:<math> [A \varphi](t) = t \varphi(t). \;</math>

Now, a physicist would say that <math>A</math> ''does'' have eigenvectors, namely the <math>\varphi(t)=\delta(t-t_0)</math>, where <math>\delta</math> is a Dirac delta-function. A delta-function, however, is not a normalizable function; that is, it is not actually in the Hilbert space {{math|''L''<sup>2</sup>[0, 1]}}. Thus, the delta-functions are "generalized eigenvectors" but not eigenvectors in the strict sense.


===Spectral subspaces and projection-valued measures===
===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, we might consider the subspace of functions supported on a small interval <math>[a,a+\epsilon]</math> inside <math>[0,1]</math>. This space is invariant under <math>A</math> and for any <math>\varphi</math> in this subspace, <math>A\varphi</math> is very close to <math>a\varphi</math>. In this approach to the spectral theorem, if <math>A</math> is a bounded self-adjoint operator, one looks for large families of such "spectral subspaces".<ref>{{harvnb|Hall|2013}} Theorem 7.2.1</ref> 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]].
In the absence of (true) eigenvectors, one can look for a "spectral subspace" consisting of an ''almost eigenvector'', i.e, a closed subspace <math>V_E</math> of <math>V</math> associated with a [[Borel set]] <math>E \subset \sigma(A)</math> in the [[Spectrum_(functional_analysis)|spectrum]] of <math>A</math>. This subspace can be thought of as the closed span of generalized eigenvectors for <math>A</math> with eigen''values'' in <math>E</math>.<ref>{{harvnb|Hall|2013}} Theorem 7.2.1</ref> In the above example, where <math> [A f](t) = t f(t), \;</math> we might consider the subspace of functions supported on a small interval <math>[a,a+\varepsilon]</math> inside <math>[0,1]</math>. This space is invariant under <math>A</math> and for any <math>f</math> in this subspace, <math>Af</math> is very close to <math>af</math>. 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 {{math|''A''}} as an integral of the coordinate function over the operator's [[Eigenvector#Infinite dimensions|spectrum]] with respect to a projection-valued measure.<ref>{{harvnb|Hall|2013}} Theorem 7.12</ref>

: <math> A = \int_{\sigma(A)} \lambda \, d E_{\lambda} .</math>

When the self-adjoint operator in question is [[compact operator|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.


One formulation of the spectral theorem expresses the operator {{math|''A''}} as an integral of the coordinate function over the operator's spectrum <math>\sigma(A)</math> with respect to a projection-valued measure.<ref>{{harvnb|Hall|2013}} Theorem 7.12</ref>
<math display="block"> A = \int_{\sigma(A)} \lambda \, d \pi (\lambda).</math>When the self-adjoint operator in question is [[compact operator|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===
===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.{{math theorem
| math_statement = Let <math>A</math> be a bounded self-adjoint operator on a Hilbert space <math> V </math>. Then there is a [[measure space]] <math>(X, \Sigma, \mu) </math> and a real-valued [[ess sup|essentially bounded]] measurable function <math>\lambda </math> on <math>X</math> and a [[unitary operator]] <math>U : V \to L^2(X, \mu)</math> such that
<math display="block"> U^* T U = A,</math>
where <math> T </math> is the [[multiplication operator]]:
<math display="block"> [T f](x) = \lambda(x) f(x) </math>
and <math> \vert T \vert </math> <math> = \vert \lambda \vert_\infty </math>.
| name = '''Theorem'''<ref>{{harvnb|Hall|2013}} Theorem 7.20</ref>
}}Multiplication operators are a direct generalization of diagonal matrices. A finite-dimensional Hermitian vector space <math>V</math> may be coordinatized as the space of functions <math>f: B \to \C </math> from a basis <math>B</math> to the complex numbers, so that the <math>B</math>-coordinates of a vector are the values of the corresponding function <math>f</math>. The finite-dimensional spectral theorem for a self-adjoint operator <math>A: V \to V </math> states that there exists an orthonormal basis of eigenvectors <math>B</math>, so that the inner product becomes the [[dot product]] with respect to the <math>B</math>-coordinates: thus <math>V</math> is isomorphic to <math>L^2( B ,\mu ) </math> for the discrete unit measure <math>\mu</math> on <math>B</math>. Also <math>A</math> is unitarily equivalent to the multiplication operator <math>[Tf](v) = \lambda(v) f(v) </math>, where <math>\lambda(v)</math> is the eigenvalue of <math>v \in B </math>: that is, <math>A</math> multiplies each <math>B</math>-coordinate by the corresponding eigenvalue <math>\lambda(v)</math>, the action of a diagonal matrix. Finally, the [[operator norm]] <math>|A| = |T| </math> is equal to the magnitude of the largest eigenvector <math>|\lambda|_\infty </math>.


The spectral theorem is the beginning of the vast research area of functional analysis called [[operator theory]]; see also [[spectral measure]].
An alternative formulation of the spectral theorem says that every bounded self-adjoint operator is unitarily equivalent to a multiplication operator. The significance of this result is that multiplication operators are in many ways easy to understand.


There is also an analogous spectral theorem for bounded [[Normal operator|normal operators]] on Hilbert spaces. The only difference in the conclusion is that now ''<math>\lambda</math>'' may be complex-valued.
'''Theorem'''.<ref>{{harvnb|Hall|2013}} Theorem 7.20</ref> Let {{math|''A''}} be a bounded self-adjoint operator on a Hilbert space {{math|''H''}}. Then there is a [[measure space]] {{math|(''X'', Σ, ''μ'')}} and a real-valued [[ess sup|essentially bounded]] measurable function {{math|''f''}} on {{math|''X''}} and a [[unitary operator]] {{math|''U'':''H'' ''L''<sup>2</sup><sub>''μ''</sub>(''X'')}} such that

::<math> U^* T U = A,</math>

:where {{math|''T''}} is the [[multiplication operator]]:

::<math> [T \varphi](x) = f(x) \varphi(x).</math>

:and <math>\|T\| = \|f\|_\infty</math>

The spectral theorem is the beginning of the vast research area of functional analysis called [[operator theory]]; see also the [[spectral measure#Spectral measure|spectral measure]].

There is also an analogous spectral theorem for bounded [[normal operator]]s on Hilbert spaces. The only difference in the conclusion is that now {{math|''f''}} may be complex-valued.


===Direct integrals===
===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.
There is also a formulation of the spectral theorem in terms of [[Direct integral|direct integrals]]. It is similar to the multiplication-operator formulation, but more canonical.


Let <math>A</math> be a bounded self-adjoint operator and let <math>\sigma (A)</math> be the spectrum of <math>A</math>. The direct-integral formulation of the spectral theorem associates two quantities to <math>A</math>. First, a measure <math>\mu</math> on <math>\sigma (A)</math>, and second, a family of Hilbert spaces <math>\{H_{\lambda}\},\,\,\lambda\in\sigma (A).</math> We then form the direct integral Hilbert space
Let <math>A</math> be a bounded self-adjoint operator and let <math>\sigma (A)</math> be the spectrum of <math>A</math>. The direct-integral formulation of the spectral theorem associates two quantities to <math>A</math>. First, a measure <math>\mu</math> on <math>\sigma (A)</math>, and second, a family of Hilbert spaces <math>\{H_{\lambda}\},\,\,\lambda\in\sigma (A).</math> We then form the direct integral Hilbert space
::<math> \int_\mathbf{R}^\oplus H_{\lambda}\, d \mu(\lambda). </math>
<math display="block"> \int_\mathbf{R}^\oplus H_{\lambda}\, d \mu(\lambda). </math>
The elements of this space are functions (or "sections") <math>s(\lambda),\,\,\lambda\in\sigma(A),</math> such that <math>s(\lambda)\in H_{\lambda}</math> for all <math>\lambda</math>.
The elements of this space are functions (or "sections") <math>s(\lambda),\,\,\lambda\in\sigma(A),</math> such that <math>s(\lambda)\in H_{\lambda}</math> for all <math>\lambda</math>.
The direct-integral version of the spectral theorem may be expressed as follows:<ref>{{harvnb|Hall|2013}} Theorem 7.19</ref>
The direct-integral version of the spectral theorem may be expressed as follows:<ref>{{harvnb|Hall|2013}} Theorem 7.19</ref>
:'''Theorem.''' If <math>A</math> is a bounded self-adjoint operator, then <math>A</math> is unitarily equivalent to the "multiplication by <math>\lambda</math>" operator on
{{math theorem|math_statement= If <math>A</math> is a bounded self-adjoint operator, then <math>A</math> is unitarily equivalent to the "multiplication by <math>\lambda</math>" operator on <math display="block"> \int_\mathbf{R}^\oplus H_{\lambda}\, d \mu(\lambda) </math>
for some measure <math>\mu</math> and some family <math>\{H_{\lambda}\}</math> of Hilbert spaces. The measure <math>\mu</math> is uniquely determined by <math>A</math> up to measure-theoretic equivalence; that is, any two measure associated to the same <math>A</math> have the same sets of measure zero. The dimensions of the Hilbert spaces <math>H_{\lambda}</math> are uniquely determined by <math>A</math> up to a set of <math>\mu</math>-measure zero.}}
::<math> \int_\mathbf{R}^\oplus H_{\lambda}\, d \mu(\lambda) </math>
for some measure <math>\mu</math> and some family <math>\{H_{\lambda}\}</math> of Hilbert spaces. The measure <math>\mu</math> is uniquely determined by <math>A</math> up to measure-theoretic equivalence; that is, any two measure associated to the same <math>A</math> have the same sets of measure zero. The dimensions of the Hilbert spaces <math>H_{\lambda}</math> are uniquely determined by <math>A</math> up to a set of <math>\mu</math>-measure zero.


The spaces <math>H_{\lambda}</math> can be thought of as something like "eigenspaces" for <math>A</math>. Note, however, that unless the one-element set <math>{\lambda}</math> has positive measure, the space <math>H_{\lambda}</math> is not actually a subspace of the direct integral. Thus, the <math>H_{\lambda}</math>'s should be thought of as "generalized eigenspace"—that is, the elements of <math>H_{\lambda}</math> are "eigenvectors" that do not actually belong to the Hilbert space.
The spaces <math>H_{\lambda}</math> can be thought of as something like "eigenspaces" for <math>A</math>. Note, however, that unless the one-element set <math>\lambda</math> has positive measure, the space <math>H_{\lambda}</math> is not actually a subspace of the direct integral. Thus, the <math>H_{\lambda}</math>'s should be thought of as "generalized eigenspace"—that is, the elements of <math>H_{\lambda}</math> 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 (the spectrum of the operator) is canonical. Second, the function we are multiplying by is canonical in the direct-integral approach: Simply the function <math>\lambda\mapsto\lambda</math>.
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 (the spectrum of the operator) is canonical. Second, the function we are multiplying by is canonical in the direct-integral approach: Simply the function <math>\lambda\mapsto\lambda</math>.


===Cyclic vectors and simple spectrum===
===Cyclic vectors and simple spectrum===
A vector <math>\varphi</math> is called a '''cyclic vector''' for <math>A</math> if the vectors <math>\varphi,A\varphi,A^2\varphi,\ldots</math> span a dense subspace of the Hilbert space. Suppose <math>A</math> is a bounded self-adjoint operator for which a cyclic vector exists. In that case, there is no distinction between the direct-integral and multiplication-operator formulations of the spectral theorem. Indeed, in that case, there is a measure <math>\mu</math> on the spectrum <math>\sigma(A)</math> of <math>A</math> such that <math>A</math> is unitarily equivalent to the "multiplication by <math>\lambda</math>" operator on <math>L^2(\sigma(A),\mu)</math>.<ref>{{harvnb|Hall|2013}} Lemma 8.11</ref> This result represents <math>A</math> simultaneously a multiplication operator ''and'' as a direct integral, since <math>L^2(\sigma(A),\mu)</math> is just a direct integral in which each Hilbert space <math>H_{\lambda}</math> is just <math>\mathbb{C}</math>.
A vector <math>\varphi</math> is called a [[cyclic vector]] for <math>A</math> if the vectors <math>\varphi,A\varphi,A^2\varphi,\ldots</math> span a dense subspace of the Hilbert space. Suppose <math>A</math> is a bounded self-adjoint operator for which a cyclic vector exists. In that case, there is no distinction between the direct-integral and multiplication-operator formulations of the spectral theorem. Indeed, in that case, there is a measure <math>\mu</math> on the spectrum <math>\sigma(A)</math> of <math>A</math> such that <math>A</math> is unitarily equivalent to the "multiplication by <math>\lambda</math>" operator on <math>L^2(\sigma(A),\mu)</math>.<ref>{{harvnb|Hall|2013}} Lemma 8.11</ref> This result represents <math>A</math> simultaneously as a multiplication operator ''and'' as a direct integral, since <math>L^2(\sigma(A),\mu)</math> is just a direct integral in which each Hilbert space <math>H_{\lambda}</math> is just <math>\mathbb{C}</math>.


Not every bounded self-adjoint operator admits a cyclic vector; indeed, by the uniqueness in the direct integral decomposition, this can occur only when all the <math>H_{\lambda}</math>'s have dimension one. When this happens, we say that <math>A</math> has "simple spectrum" in the sense of [[Self-adjoint_operator#Spectral_multiplicity_theory|spectral multiplicity theory]]. That is, a bounded self-adjoint operator that admits a cyclic vector should be thought of as the infinite-dimensional generalization of a self-adjoint matrix with distinct eigenvalues (i.e., each eigenvalue has multiplicity one).
Not every bounded self-adjoint operator admits a cyclic vector; indeed, by the uniqueness in the direct integral decomposition, this can occur only when all the <math>H_{\lambda}</math>'s have dimension one. When this happens, we say that <math>A</math> has "simple spectrum" in the sense of [[Self-adjoint operator#Spectral multiplicity theory|spectral multiplicity theory]]. That is, a bounded self-adjoint operator that admits a cyclic vector should be thought of as the infinite-dimensional generalization of a self-adjoint matrix with distinct eigenvalues (i.e., each eigenvalue has multiplicity one).


Although not every <math>A</math> 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 <math>A</math> has a cyclic vector. This observation is the key to the proofs of the multiplication-operator and direct-integral forms of the spectral theorem.
Although not every <math>A</math> 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 <math>A</math> has a cyclic vector. This observation is the key to the proofs of the multiplication-operator and direct-integral forms of the spectral theorem.


===Functional calculus===
===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 <math>f</math> defined on the spectrum of <math>A</math>, we wish to define an operator <math>f(A)</math>. If <math>f</math> is simply a positive power, <math>f(x)=x^n</math>, then <math>f(A)</math> is just the <math>n\mathrm{th}</math> power of <math>A</math>, <math>A^n</math>. The interesting cases are where <math>f</math> 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.<ref>E.g., {{harvnb|Hall|2013}} Definition 7.13</ref> In the direct-integral version, for example, <math>f(A)</math> acts as the "multiplication by <math>f</math>" operator in the direct integral:
One important application of the spectral theorem (in whatever form) is the idea of defining a [[functional calculus]]. That is, given a function <math>f</math> defined on the spectrum of <math>A</math>, we wish to define an operator <math>f(A)</math>. If <math>f</math> is simply a positive power, <math>f(x) = x^n</math>, then <math>f(A)</math> is just the <math>n</math>-th power of <math>A</math>, <math>A^n</math>. The interesting cases are where <math>f</math> 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.<ref>E.g., {{harvnb|Hall|2013}} Definition 7.13</ref> In the direct-integral version, for example, <math>f(A)</math> acts as the "multiplication by <math>f</math>" operator in the direct integral:
:<math>[f(A)s](\lambda)=f(\lambda)s(\lambda)</math>.
<math display="block">[f(A)s](\lambda) = f(\lambda) s(\lambda).</math>
That is to say, each space <math>H_{\lambda}</math> in the direct integral is a (generalized) eigenspace for <math>f(A)</math> with eigenvalue <math>f(\lambda)</math>.
That is to say, each space <math>H_{\lambda}</math> in the direct integral is a (generalized) eigenspace for <math>f(A)</math> with eigenvalue <math>f(\lambda)</math>.


== General self-adjoint operators ==
== Unbounded self-adjoint operators ==
Many important linear operators which occur in [[Mathematical analysis|analysis]], such as [[differential operators]], are unbounded. There is also a spectral theorem for [[self-adjoint operator]]s that applies in these cases. To give an example, every 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 [[Multiplier (Fourier analysis)|Fourier multiplier]].
Many important linear operators which occur in [[Mathematical analysis|analysis]], such as [[differential operators]], are [[unbounded operator|unbounded]]. There is also a spectral theorem for [[self-adjoint operator]]s that applies in these cases. To give an example, every 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 [[Multiplier (Fourier analysis)|Fourier multiplier]].


In general, spectral theorem for self-adjoint operators may take several equivalent forms.<ref>See Section 10.1 of {{harvnb|Hall|2013}}</ref> Notably, all of the formulations given in the previous section for bounded self-adjoint operators—the projection-valued measure version, the multiplication-operator version, and the direct-integral version—continue to hold for unbounded self-adjoint operators, with small technical modifications to deal with domain issues.
In general, spectral theorem for self-adjoint operators may take several equivalent forms.<ref>See Section 10.1 of {{harvnb|Hall|2013}}</ref> Notably, all of the formulations given in the previous section for bounded self-adjoint operators—the projection-valued measure version, the multiplication-operator version, and the direct-integral version—continue to hold for unbounded self-adjoint operators, with small technical modifications to deal with domain issues. Specifically, the only reason the multiplication operator <math>A</math> on <math>L^2([0,1])</math> is bounded, is due to the choice of domain <math>[0,1]</math>. The same operator on, e.g., <math>L^2(\mathbb{R})</math> would be unbounded.

The notion of "generalized eigenvectors" naturally extends to unbounded self-adjoint operators, as they are characterized as [[Probability_amplitude#Normalization|non-normalizable]] eigenvectors. Contrary to the case of [[Spectral_theorem#Spectral_subspaces_and_projection-valued_measures|almost eigenvectors]], however, the eigenvalues can be real or complex and, even if they are real, do not necessarily belong to the spectrum. Though, for self-adjoint operators there always exist a real subset of "generalized eigenvalues" such that the corresponding set of eigenvectors is [[Total_set|complete]].{{sfn|de la Madrid Modino|2001|pp=95-97}}


== See also ==
== See also ==
* {{annotated link|Hahn-Hellinger theorem}}
* [[Spectral theory of compact operators]]
* [[Spectral theory of normal C*-algebras]]
* [[Borel functional calculus]]
* [[Borel functional calculus]]
* [[Spectral theory]]
* [[Spectral theory]]
Line 140: Line 134:
* [[Singular value decomposition]], a generalisation of spectral theorem to arbitrary matrices.
* [[Singular value decomposition]], a generalisation of spectral theorem to arbitrary matrices.
* [[Eigendecomposition of a matrix]]
* [[Eigendecomposition of a matrix]]
* [[Wiener–Khinchin theorem]]


== Notes ==
== Notes ==
{{reflist}}
{{reflist}}


== References ==
==References==
{{Reflist}}
* [[Sheldon Axler]], ''Linear Algebra Done Right'', Springer Verlag, 1997
* [[Sheldon Axler]], ''Linear Algebra Done Right'', Springer Verlag, 1997
*{{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267 | year = 2013 |publisher = Springer|isbn=978-1461471158}}
* {{citation | last = Hall |first = B.C. |title = Quantum Theory for Mathematicians|series=Graduate Texts in Mathematics|volume=267 | year = 2013 |publisher = Springer|bibcode = 2013qtm..book.....H |isbn=978-1461471158}}
* [[Paul Halmos]], [https://www.jstor.org/stable/2313117 "What Does the Spectral Theorem Say?"], ''American Mathematical Monthly'', volume 70, number 3 (1963), pages 241–247 [http://www.math.wsu.edu/faculty/watkins/Math502/pdfiles/spectral.pdf Other link]
* [[Paul Halmos]], [https://www.jstor.org/stable/2313117 "What Does the Spectral Theorem Say?"], ''American Mathematical Monthly'', volume 70, number 3 (1963), pages 241–247 [http://www.math.wsu.edu/faculty/watkins/Math502/pdfiles/spectral.pdf Other link]
*{{cite thesis |last=de la Madrid Modino |first= R. |date=2001 |title= Quantum mechanics in rigged Hilbert space language|url=https://scholar.google.com/scholar?oi=bibs&cluster=2442809273695897641&btnI=1&hl=en |degree= PhD |publisher= Universidad de Valladolid}}
* [[Michael C. Reed|M. Reed]] and [[Barry Simon|B. Simon]], ''Methods of Mathematical Physics'', vols I–IV, Academic Press 1972.
* [[Michael C. Reed|M. Reed]] and [[Barry Simon|B. Simon]], ''Methods of Mathematical Physics'', vols I–IV, Academic Press 1972.
* [[Gerald Teschl|G. Teschl]], ''Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators'', http://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
* [[Gerald Teschl|G. Teschl]], ''Mathematical Methods in Quantum Mechanics with Applications to Schrödinger Operators'', https://www.mat.univie.ac.at/~gerald/ftp/book-schroe/, American Mathematical Society, 2009.
* {{Cite book |title=Spectral Theory and Quantum Mechanics; Mathematical Foundations of Quantum Theories, Symmetries and Introduction to the Algebraic Formulation 2nd Edition |author= Valter Moretti |author-link= Valter Moretti |publisher= Springer |year=2017 |url=https://www.springer.com/it/book/9783319707051|isbn=978-3-319-70705-1 }}



{{Functional Analysis}}
{{Functional analysis}}
{{Spectral theory}}


[[Category:Spectral theory|*]]
[[Category:Spectral theory|*]]

Latest revision as of 00:05, 13 December 2024

In 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 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.[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 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

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Hermitian maps and Hermitian matrices

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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 finite-dimensional complex inner product space V endowed with a positive definite sesquilinear inner product The Hermitian condition on means that for all x, yV,

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* is equal to its conjugate transpose. (If A is a real matrix, then this is equivalent to AT = A, that is, A is a symmetric matrix.)

This condition implies that all eigenvalues of a Hermitian map are real: To see this, 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 non-zero vector v such that A v = λv for some scalar λ. The value λ is the corresponding eigenvalue. Moreover, the eigenvalues are roots of the characteristic polynomial.)

Theorem — If A is Hermitian on V, then there exists an orthonormal basis of V consisting of eigenvectors of A. Each eigenvalue of A 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 complex eigenvalue λ1 and corresponding eigenvector v1 , which must by definition be non-zero. Then since we find that λ1 is real. Now consider the space the orthogonal complement of v1 . By Hermiticity, is an invariant subspace of A. 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 A 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 A 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. A 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, V 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 (see the proof in case of normal matrices below).

Spectral decomposition and the singular value decomposition

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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

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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 using the Schur decomposition. That is, any matrix can be written as A = U T U* , where U is unitary and T is upper triangular. If A is normal, then one sees that T T* = 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 self-adjoint operators

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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.

Theorem — Suppose A is a compact self-adjoint 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, 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

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Possible absence of eigenvectors

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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 A be the operator of multiplication by t on , that is,[3]

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 L2[0, 1] 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

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In the absence of (true) 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 .[4] 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 A as an integral of the coordinate function over the operator's spectrum with respect to a projection-valued measure.[5] 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

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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.

Theorem[6] — Let be a bounded self-adjoint operator on a Hilbert space . Then there is a measure space and a real-valued essentially bounded measurable function on and a unitary operator such that where is the multiplication operator: and .

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

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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 (or "sections") such that for all . The direct-integral version of the spectral theorem may be expressed as follows:[7]

Theorem — If is a bounded self-adjoint 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 measure-theoretic 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 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 (the spectrum of the operator) is canonical. Second, the function we are multiplying by is canonical in the direct-integral approach: Simply the function .

Cyclic vectors and simple spectrum

[edit]

A vector is called a cyclic vector for if the vectors span a dense subspace of the Hilbert space. Suppose is a bounded self-adjoint operator for which a cyclic vector exists. In that case, there is no distinction between the direct-integral and multiplication-operator 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 as a multiplication operator and as a direct integral, since is just a direct integral in which each Hilbert space is just .

Not every bounded self-adjoint 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 self-adjoint operator that admits a cyclic vector should be thought of as the infinite-dimensional generalization of a self-adjoint 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 multiplication-operator and direct-integral forms of the spectral theorem.

Functional calculus

[edit]

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 -th 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 direct-integral 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 .

Unbounded self-adjoint operators

[edit]

Many important linear operators which occur in analysis, such as differential operators, are unbounded. There is also a spectral theorem for self-adjoint operators that applies in these cases. To give an example, every 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.

In general, spectral theorem for self-adjoint operators may take several equivalent forms.[10] Notably, all of the formulations given in the previous section for bounded self-adjoint operators—the projection-valued measure version, the multiplication-operator version, and the direct-integral version—continue to hold for unbounded self-adjoint operators, with small technical modifications to deal with domain issues. Specifically, the only reason the multiplication operator on is bounded, is due to the choice of domain . The same operator on, e.g., would be unbounded.

The notion of "generalized eigenvectors" naturally extends to unbounded self-adjoint operators, as they are characterized as non-normalizable eigenvectors. Contrary to the case of almost eigenvectors, however, the eigenvalues can be real or complex and, even if they are real, do not necessarily belong to the spectrum. Though, for self-adjoint operators there always exist a real subset of "generalized eigenvalues" such that the corresponding set of eigenvectors is complete.[11]

See also

[edit]

Notes

[edit]
  1. ^ Hawkins, Thomas (1975). "Cauchy and the spectral theory of matrices". Historia Mathematica. 2: 1–29. doi:10.1016/0315-0860(75)90032-4.
  2. ^ A Short History of Operator Theory by Evans M. Harrell II
  3. ^ Hall 2013 Section 6.1
  4. ^ Hall 2013 Theorem 7.2.1
  5. ^ Hall 2013 Theorem 7.12
  6. ^ Hall 2013 Theorem 7.20
  7. ^ Hall 2013 Theorem 7.19
  8. ^ Hall 2013 Lemma 8.11
  9. ^ E.g., Hall 2013 Definition 7.13
  10. ^ See Section 10.1 of Hall 2013
  11. ^ de la Madrid Modino 2001, pp. 95–97.

References

[edit]