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Topic: Spectral theorem

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List of functional analysis topics

Spectral theory

Spectrum of an operator

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Differential operator

Invariant subspace

Eigenfunction

Hilbert space

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Eigenvalue

Conjugate transpose

Inner product space

Diagonal matrix

Gelfand representation

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	NationMaster - Encyclopedia: Spectral theorem
		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 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 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.
	www.nationmaster.com /encyclopedia/Spectral-theorem (3217 words)

	Spectral theorem - Wikipedia, the free encyclopedia
		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 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 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.
	en.wikipedia.org /wiki/Spectral_theorem (1391 words)

	NationMaster - Encyclopedia: Spectral theory
		In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix.
		The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting.
		In Mathematics it is Spectral Geometry, which links spectral properties of elliptic operators and related properties of parabolic operators to the geometry and topology of the underlying manifold.
	www.nationmaster.com /encyclopedia/Spectral-theory (1303 words)

	PlanetMath: spectral theorem
		There are several versions of increasing sophistication of the spectral theorem that hold in infinite-dimensional, Hilbert space setting.
		Finally, there are versions of the spectral theorem, of importance in theoretical quantum mechanics, that can be applied to continuous 1-parameter groups of commuting, self-adjoint operators.
		This is version 4 of spectral theorem, born on 2002-06-07, modified 2002-12-01.
	planetmath.org /encyclopedia/SpectralTheoremForHermitianMatrices.html (262 words)

	Spectral theory - Wikipedia, the free encyclopedia
		In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix.
		The original spectral theorem was therefore conceived as a version of the theorem on principal axes of an ellipsoid, in an infinite-dimensional setting.
		After Hilbert's initial formulation, the later development of abstract Hilbert space and the spectral theory of a single normal operator on it did very much go in parallel with the requirements of physics; particularly at the hands of von Neumann.
	en.wikipedia.org /wiki/Spectral_theory (336 words)

	Wikinfo \| Spectral theorem (Site not responding. Last check: 2007-11-05)
		In mathematics, the spectral theorem is an important decomposition theorem applying to normal operators in linear algebra and functional analysis.
		There is also a spectral theorem for normal operators on Hilbert spaces, though, in which the sum in the finite-dimensional spectral theorem is replaced by an integral of the coordinate function over the spectrum against a projection-valued measure.
		Jordan decomposition, an "algebraic" analogue to spectral decomposition.
	www.wikinfo.org /wiki.php?title=Spectral_theorem (1065 words)

	PlanetMath: spectral theorem
		There are several versions of increasing sophistication of the spectral theorem that hold in infinite-dimensional, Hilbert space setting.
		Finally, there are versions of the spectral theorem, of importance in theoretical quantum mechanics, that can be applied to continuous 1-parameter groups of commuting, self-adjoint operators.
		This is version 4 of spectral theorem, born on 2002-06-07, modified 2002-12-01.
	www.planetmath.org /encyclopedia/SpectralTheoremForHermitianMatrices.html (262 words)

	AMERICAN MATHEMATICAL MONTHLY -February 2001
		Several linear relationships are obtained for the areas of the squares in consecutive bands; in particular, we generalize a classical theorem of Euler from quadrilaterals to general polygons.
		The definitions of the adjoint and of the notion of normality are made with the aid of this theorem.
		Finally the spectral theorem for normal linear transformations is proved, both in the complex and the real case.
	www.maa.org /pubs/monthly_feb_01_toc.html (580 words)

	Spectral theorem
		The spectral theorem is also true for symmetric operators on finite-dimensional real inner product spaces.
		There is however a spectral theorem for self-adjoint operators which applies in many of these cases.
		Jordan decomposition, an "algebraic" analogue to spectral decomposition.
	www.brainyencyclopedia.com /encyclopedia/s/sp/spectral_theorem.html (1211 words)

	MERL – TR2002-042 – A unifying theorem for spectral embedding and clustering (Site not responding. Last check: 2007-11-05)
		MERL – TR2002-042 – A unifying theorem for spectral embedding and clustering
		Spectral methods use selected eigenvectors of a data affinity matrix to obtain a data representation that can be trivially clustered or embedded in a low-dimensional space.
		We present a theorem that explains, for broad classes of affinity matrices and eigenbases, why this works: For successively smaller eigenbases (i.e., using fewer and fewer of the affinity matrix's dominant eigenvalues and eigenvectors), the angles between similar vectors in the new representation shrink while the angles between dissimilar vectors grow.
	www.merl.com /papers/TR2002-42 (230 words)

	baudline manual - glossary
		Constant spectral energy at all frequencies with a probability histogram that follows a Gaussian bell shaped curve.
		The amount to move the FFT sliding window for smooth and continuous spectral slice processing.
		QAM has better data throughput and better spectral efficiency than PSK but it requires a higher SNR to achieve an equivalent BER.
	www.baudline.com /manual/glossary.html (2016 words)

	FuncAna
		Converse of the spectral theorem: every operator that admits the spectral decomposition with real eigenvalues is compact and self-adjoint.
		Statement of the spectral decomposition in terms of spectral projectors (orthogonal projectors on eigenspaces).
		Its poles are the eigenvalues and the corresponding residues are the spectral projectors.
	www.math.ttu.edu /~vshubov/FuncAna/FuncAna.html (947 words)

	SVD
		Theorem: If A is an m x n matrix, then there is an orthonormal basis of
		If you haven't seen the spectral theorem, then skip this proof.
		Theorem: Every matrix has a singular value decomposition.
	www.uwlax.edu /faculty/will/svd/svd/index.html (587 words)

	Spectral radius Information
		In mathematics, the spectral radius of a matrix or a bounded linear operator is the supremum among the moduli of the elements in its spectrum, which is sometimes denoted by ρ(·).
		The spectral radius is closely related to the behaviour of the convergence of the power sequence of a matrix; namely, the following theorem holds:
		The spectral radius of a planar graph is defined to be the spectral radius of its adjacency matrix.
	www.bookrags.com /wiki/Spectral_radius (421 words)

	[No title]
		The purpose of the present paper, however, is to explain the proof of Thoma- son's Theorem, and the vast array of machinery underlying it.
		The analogue for spectra of a theorem of Tate.
		Theorem 1.1 Let F be a separably closed field, ` a prime not equal to the characteristic of F. Then ss2nKF ^~= Z`(n) for n 0, and ssoddKF ^= 0.
	hopf.math.purdue.edu /Mitchell/thomason.txt (10189 words)

	Department of Mathematics - University of Georgia
		Beginning with a careful study of integers, modular arithmetic, the Euclidean algorithm, the course moves on to fields, isometries of the complex plain, polynomials, splitting fields, rings, homomorphisms, field extensions and compass and straightedge constructions.
		Topics include the finite-dimensional spectral theorem, group actions, classification of finitely generated modules over principal ideal domains, and canonical forms of linear operators.
		Hahn, Jordan and Lebesgue decomposition theorems, Radon-Nikodym Theorem and Fubini's Theorem.
	www.math.uga.edu /graduate/GraduateCourses.html (1895 words)

	Bandwidth Definitions, Bandwidth Resources, Bandwidth Quotes, Bandwidth T1, T3
		In radio communications, for example, bandwidth is the range of frequencies occupied by a modulated carrier wave, whereas in optics it is the width of an individual spectral line or the entire spectral range.
		According to the Shannon—Hartley theorem, the data rate of reliable communication is directly proportional to the frequency range of the signal used for the communication.
		When Additive white Gaussian noise is present in a digital communication channel, the Shannon—Hartley theorem gives the relationship between the channel's bandwidth, the channel's capacity, and the Signal-to-noise ratio (SNR) ratio of the system.
	www.solveforce.com /bandwidth.html (1443 words)

	Description of Research - Injectivity of maps
		In [15] we use Pinchuk's counterexample to the real Jacobian conjecture to construct certain local diffeomorphisms which show that there is no higher dimensional version of the spectral theorem of Gutierrez.
		On the positive side, we prove in [15] a sharp injectivity theorem with "nearly spectral" hypothesis, similar in spirit to the one of Gutierrez that is valid in all dimensions.
		In [16] we extended a special but important case of the theorem of Gutierrez by geometrizing it to complete surfaces of non-positive curvature.
	www.nd.edu /~fxavier/Research/injectivity.htm (573 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		In Section~\ref{s3} we introduce the spectral shift operator $\Xi(\lambda,H_0,H)$ associated with the pair $(H_0,H)$ and relate it to Krein's spectral shift function $\xi(\lambda,H_0,H)$ and his celebrated trace formula \cite{Kr62}.
		\le (\pi / 2),$ $\mu\in\bbR,$ $\varepsilon > 0$ by the spectral theorem for the self-adjoint operator $L$, we infer that the family of operators $\int_\varepsilon^\infty d\lambda \, L(L^2+\lambda^2)^{-1}$ is uniformly bounded in $\varepsilon >0$ and therefore, it suffices to check the convergence \eqref{n2.30} on a dense set in $\calK$.
		% \bi{Si75} K.~B.~Sinha, {\it On the theorem of M.~G.~Krein, } preprint, 1975, unpublished.
	www.ma.utexas.edu /mp_arc/papers/99-30 (4701 words)

	MATH 502 Spring 1997, Lecture Schedule (Site not responding. Last check: 2007-11-05)
		Spectral Theorem for compact self-adjoint operators in Hilbert space
		Spectral Theorem for compact normal operators in Hilbert space
		Spectral Mapping Theorem; Spectral Theorem for self-adjoint operators in Hilbert space
	www.ima.umn.edu /~arnold/502.s97/functionallectures.html (215 words)

	Spectral Clustering, ICML 2004 Tutorial by Chris Ding
		At the core of spectral clustering is the Laplacian of the graph adjacency (pairwise similarity) matrix, evolved from spectral graph partitioning.
		A unifying theorem for spectral embedding and clustering.
		Spectral relaxation models and structure analysis for k-way graph Clustering and bi-clustering.
	crd.lbl.gov /~cding/Spectral (721 words)

	Math Forum Discussions
		Re: Is there a spectral theorem mod 2 (i.e.
		Is there a spectral theorem mod 2 (i.e.
		The Math Forum is a research and educational enterprise of the Drexel School of Education.
	mathforum.org /kb/thread.jspa?messageID=529352&tstart=0 (137 words)

	Department of Mathematics, University of Strathclyde
		General aims: To extend the ideas and applications of spectral theory, familiar when solving algebraic equations, to problems on infinite-dimensional spaces involving linear operators which are not necessarily bounded.
		Spectral Theory of Bounded Linear Operators: The spectrum sigma(T) and its components.
		Statement of the Spectral Theorem on infinite-dimensional spaces.
	www.maths.strath.ac.uk /ungrad/classes/921.htm (267 words)

	[No title] (Site not responding. Last check: 2007-11-05)
		BANACH SPACES Bounded linear maps (Hahn-Banach theorems, open mapping theorem, closed graph theorem), Hilbert spaces (Riesz lemma, Hilbert space bases); adjoint maps (definition and basic properties, basic properties of normal, unitary and projection operators); complexification of real Banach spaces.
		SPECTRAL THEORY Spectra in Banach algebras (nonemptiness, the spectral radius formula, the functional calculus, the spectral mapping theorem); different types of spectral points in algebras of linear operators; the spectral theorem for compact operators; normal operators in Hilbert space (spectral measures and the spectral theorem).
		CURVATURE Definition and interpretation; Gaussian curvature; the second fundamental form; Gauss' theorem on sectional curvature of immersed manifolds; totally geodesic immersions; why Gaussian curvature in two dimensions is especially important.
	www.math.rutgers.edu /grad/oral/bk.txt (261 words)

	Vasco Brattka's Papers (Site not responding. Last check: 2007-11-05)
		Computing the spectral decomposition of a normal matrix is among the most frequent tasks to numerical mathematics.
		We investigate the spectral representation's effectivity properties on the sound formal basis of computable analysis.
		Thus, in principle the spectral decomposition can be computed under remarkably weak non-degeneracy conditions.
	www.informatik.fernuni-hagen.de /import/thi1/vasco.brattka/publications/spectral.html (132 words)

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