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Topic: Hermitian

Related Topics

Hermitian conjugate

Hermitian matrix

Conjugate transpose

List of matrices

Normal matrix

Symmetric matrix

Diagonal matrix

Unitary matrix

Inverse matrix

Matrix (mathematics)

Stochastic matrix

Toeplitz matrix

Complex conjugate

Matrix inversion

Singular value decomposition

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	PlanetMath: Hermitian matrix (Site not responding. Last check: 2007-10-07)
		Hermitian matrices is a vector subspace in the vector space of complex matrices.
		Hermitian matrices are named after Charles Hermite (1822-1901) [2], who proved in 1855 that the eigenvalues of these matrices are always real [1].
		This is version 10 of Hermitian matrix, born on 2002-01-21, modified 2005-05-20.
	planetmath.org /encyclopedia/HermitianMatrix.html (189 words)

	Hermitian (Site not responding. Last check: 2007-10-07)
		It says that any Hermitian matrix can be diagonalized by a unitary matrix, and that the resulting diagonal matrix has only real entries.
		A version of the spectral theorem also applies to Hermitian operators; while the eigenvectors to different eigenvalues are orthogonal, in general it is not true that the Hilbert space H admits an orthonormal basis consisting only of eigenvectors of the operator.
		For this reason, one considers even more general Hermitian operators in the mathematical formulation of quantum mechanics : these operators are only defined on a dense subspace of a Hilbert space and don't have to be continuous.
	www.serebella.com /encyclopedia/article-Hermitian.html (581 words)

	Hermitian matrix - Wikipedia, the free encyclopedia
		A Hermitian matrix (or self-adjoint matrix) is a square matrix with complex entries which is equal to its own conjugate transpose — that is, the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:
		A matrix that has only real entries is Hermitian if and only if it is a symmetric matrix, i.e., if it is symmetric with respect to the main diagonal.
		If the eigenvalues of a Hermitian matrix are all positive, then the matrix is positive definite; if they are all non-negative, then the matrix is positive semidefinite.
	en.wikipedia.org /wiki/Hermitian_matrix (402 words)

	Hermitian Matrices
		A Hermitian matrix is defined as a matrix that is equal to its Hermitian conjugate.
		Returning to Hermitian matrices, we can also show that they possess another very interesting property, namely, that their eigenvectors are mutually orthogonal (assuming distinct eigenvalues) in a sense to be defined below.
		Hermitian matrices have found an important application in modern physics, as the representations of measurement operators in Heisenberg's version of quantum mechanics.
	www.mathpages.com /home/kmath306/kmath306.htm (744 words)

	Hermitian adjoint
		The Hermitian adjoint can be generally defined using only the axioms (properties) of Hilbert space and the inner product.
		Hermitian operators are very important because of their properties.
		The hermitian adjoint of a complex matrix is the transpose of the matrix of the complex conjugates of the elements of the original matrix:
	en.mcfly.org /Hermitian+adjoint (328 words)

	hermitian (Site not responding. Last check: 2007-10-07)
		In this article we discuss the term Hermitian as used in operator and matrix theory to refer to a certain kind of operator (or matrix).
		This means that all eigenvaluess of a Hermitian matrix are real, and, moreover, eigenvectors with distinct eigenvalues are orthogonal.
		Matrix theorists sometimes refer to real Hermitian matrices as symmetric matrices, since indeed they are symmetric with respect to the diagonal.
	www.yourencyclopedia.net /Hermitian.html (239 words)

	Hermitian Adjoint (Site not responding. Last check: 2007-10-07)
		In mathematics, the hermitian adjoint of an linear operator is a matching operator (very similar to the inverse operator in concept) defined over a linear space with inner product.
		The Hamiltonian (energy) of physical systems is the most important Hermitian operator, since its govern the developement of the system with time.
		The concept is generalized from matrices of (real and/or complex) numbers to linear operators generally, and from vector spaces to function spaces.
	www.wikiverse.org /hermitian-adjoint (336 words)

	Hermitian K-theory of the integers, by A. J. Berrick and M. Karoubi (Site not responding. Last check: 2007-10-07)
		Hermitian K-theory of the integers, by A. Berrick and M. Karoubi
		In this paper we prove analogous results for the Hermitian K-theory of the integers with 2 inverted (denoted by Z').
		We also prove in this case the analog of the Lichtenbaum conjecture for the Hermitian K-theory of Z' : the homotopy fixed point set of a suitable Z/2 action on the classifying space of the algebraic K-theory of Z' is the Hermitian K-theory of Z' after 2-adic completion.
	www.math.uiuc.edu /K-theory/0649 (108 words)

	Maintaining the Hermitian Property
		Another way to look at the Hermitian property is to explore the real and imaginary component independently (see Figure 5.2).
		How the Hermitian relation applies to the indexing scheme in the discrete case is presented in Figure 5.3, there are two different patterns for even and odd number of samples.
		That means all operations applied in the frequency domain have to preserve the Hermitian property in order to obtain a real function after the inverse Fourier transform.
	www.cg.tuwien.ac.at /courses/projekte/vis/finished/MArtner/node50.html (492 words)

	Hermitian Operators (Site not responding. Last check: 2007-10-07)
		Hermitian operators on quantum general relativity loop space...
		A Hermitian operator conjugate to the number operator...
		Remarks on the construction of a Hermitian phase operator...
	www.scienceoxygen.com /phys/151.html (118 words)

	Hermitian (Site not responding. Last check: 2007-10-07)
		A Hermitian matrix is a square matrix with complex entries so that the matrix is equal to its own conjugate transpose - that is, if the element in the ith rowand jth column is equal to the complex conjugate of theelement in the jth row and ith column, for all indices i and j:
		If all the entries of a matrix are real, then it is Hermitian if and onlyif it is a symmetric matrix.
		A version of the spectral theorem also applies to Hermitian operators; while the eigenvectors corresponding to differenteigenvalues are orthogonal, in general it is not true that the Hilbert space H admits an orthonormal basisconsisting only of eigenvectors of the operator.
	www.therfcc.org /hermitian-45319.html (469 words)

	Hermitian
		In mathematics, a square matrix with complex entries is called Hermitian if it is equal to its conjugate transpose - that is, if the element in the ith row and jth column is equal to the complex conjugate of the element in the jth row and ith column, for all indices i and j:
		In the mathematical formulation of quantum mechanics, one considers even more general Hermitian operators: they are only defined on a dense subspace of a Hilbert space and don't have to be continuous.
		They swarmed numerous like locusts, blind and orderly and absorbed, impervious to sentiment, to logic, Often while walking abroad, when he happened also to come out of mankind.
	www.wordlookup.net /he/hermitian.html (730 words)

	Introduction
		This assumption is true for a wide class of practically important cases, and the theory is very closely related to the standard Hermitian eigenproblem, as expounded in Chapter 4.
		For a Hermitian definite pencil, the reducing subspace is of the same dimension as the multiplicity of the eigenvalue.
		It is important to keep in mind that when a couple of eigenvalues coincide, their eigenvectors lose their individuality: there is no way of saying that one set of vectors comprises the eigenvectors of a multiple eigenvalue.
	www.cs.utk.edu /~dongarra/etemplates/node156.html (316 words)

	Hermitian Operators
		Another important property of Hermitian operators is that their eigenvectors are orthogonal (or can be chosen to be so).
		Thus we have shown that eigenfunctions of a Hermitian operator with different eigenvalues are orthogonal.
		This Schmidt-orthogonalization procedure can be extended to the case of n-fold degeneracy, so we have shown that for a Hermitian operator, the eigenvectors can be made orthogonal.
	vergil.chemistry.gatech.edu /notes/quantrev/node16.html (194 words)

	Quantum Mechanics
		In standard texts on quantum mechanics, the vector is represented by a function known as the wave-function, or ψ-function.
		Hermitian operators in the Hilbert space associated with a system represent physical quantities, and their eigenvalues represent the possible results of measurements of those quantities.
		(P2) For any Hermitian operator on a Hilbert space, there are others, on the same space, with which it doesn't share a full set of eigenvectors; indeed, it is easy to show that there are other such operators with which it has no eigenvectors in common.
	plato.stanford.edu /entries/qm (3833 words)

	Notes 6
		We talked earlier about Hermitian operators, we defined them before to be operators whose expectation values are always real.
		There is a very different definition of Hermitian operators, a more basic one, and we will see the connection between the two definitions later (we'll prove that they're really equivalent).
		Hermitian operators are very special, and as we've already stated, they play a key role in quantum mechanics.
	www.colorado.edu /physics/phys3220/3220_fa97/notes/notes6/3220_notes6_3.html (915 words)

	NMath Matrix User's Guide - 2.4 Hermitian Matrices
		A Hermitian matrix is a square matrix which satisfies
		According to the strict definition of a Hermitian matrix, the diagonal elements must be real numbers, since
		A symmetric matrix is a special case of a Hermitian matrix where all the elements are real (Section 2.3).
	www.centerspace.net /doc/NMath/Matrix/user/matrixtypes5.html (141 words)

	Regular Extensions of Hermitian Operators
		The concept of regular extensions of an Hermitian (non-densely defined) operator was introduced by A. Kuzhel in 1980.
		Chapter 2 is devoted to the investigation of regular extensions of Hermitian (symmetric) operators with certain restrictions.
		In chapter 3 regular extensions of Hermitian operators with the use of boundary-value spaces are investigated.
	www.vsppub.com /books/mathe/bk-RegExtHerOpe.html (231 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		SPPSVX Solves a symmetric or Hermitian positive definite system of CPPSVX linear equations AX = B (A is held in packed storage) and provides an estimate of the condition number and error bounds on the solution.
		CHPSVX Solves a complex Hermitian indefinite system of linear equations AX = B (A is held in packed storage) and provides an estimate of the condition number and error bounds on the solution.
		CHPTRS Solves a complex Hermitian indefinite system of linear equations AX = B (A is held in packed storage) using the factorization computed by CHPTRF.
	www.sdsc.edu /SDSCwire/Services/c90_opt_08_27_97/lapack.txt (8350 words)

	check_hermitian (Site not responding. Last check: 2007-10-07)
		The function check_hermitian is used to check whether a matrix is Hermitian or non-Hermitian.
		If the input argument h is _hermitian, then the matrix elements are checked and, if possible, the matrix is converted to a hermitian matrix.
		A flag whether a is nor hermitian or not.
	gamma.magnet.fsu.edu /html/modules/matrix/matrix5.htm (242 words)

	NMath Matrix User's Guide - 2.8 Hermitian Banded Matrices
		A Hermitian banded matrix is a Hermitian matrix (Section 2.4) that has all its non-zero entries near the diagonal.
		A symmetric banded matrix is a special case of a Hermitian banded matrix where all the elements are real (Section 2.7).
		For efficiency, the lower triangular part of the matrix and zero elements outside the bandwidth are not stored.
	www.centerspace.net /doc/NMath/Matrix/user/matrixtypes9.html (185 words)

	Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems - Bai, Golub, Ng ... (Site not responding. Last check: 2007-10-07)
		Abstract: We study ecient iterative methods for the large sparse non-Hermitian positive de nite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coecient matrix.
		These methods include a Hermitian/skew-Hermitian splitting (HSS) iteration and its inexact variant, the inexact Hermitian/skew-Hermitian splitting (IHSS) iteration, which employs some Krylov subspace methods as its inner iteration processes at each step of the outer HSS iteration.
		Bai, G. Golub, and M. Ng, Hermitian and skew-Hermitian splitting methods for non-Hermitian positive definite linear systems, Technical Report SCCM-01-06, Scientific Computing and Computational Mathematics Program, Department of Computer Science, Stanford University, 2001.
	citeseer.ist.psu.edu /bai01hermitian.html (562 words)

	Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems
		Hermitian and Skew-Hermitian Splitting Methods for Non-Hermitian Positive Definite Linear Systems: SIAM Journal on Matrix Analysis and Applications Vol.
		We study efficient iterative methods for the large sparse non-Hermitian positive definite system of linear equations based on the Hermitian and skew-Hermitian splitting of the coefficient matrix.
		Moreover, we derive an upper bound of the contraction factor of the HSS iteration which is dependent solely on the spectrum of the Hermitian part and is independent of the eigenvectors of the matrices involved.
	epubs.siam.org /sam-bin/dbq/article/39545 (243 words)

	Hermitian (Site not responding. Last check: 2007-10-07)
		In this article we discuss the Hermitian as used in operator and matrix to refer to a certain kind of (or matrix).
		It says that any Hermitian can be diagonalized by a unitary matrix and that the resulting diagonal matrix only real entries.
		Matrix theorists sometimes refer to real matrices as symmetric matrices since indeed they are symmetric respect to the diagonal.
	www.freeglossary.com /Hermitian (271 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		If more than one routine name appears for a given block in the table, the first name listed is the name of the man page that documents all routines listed in that block.
		The table is in alphabetic order, except that each Hermitian matrix routine (any routine whose name begins with CH) is grouped next to equivalent symmetric matrix routines (whose names begin with SS or CS).
		This is because the Hermitian property is a type of symmetry.
	www.sdsc.edu /SDSCwire/Services/c90_opt_08_27_97/blas2.txt (894 words)

	Hermitian operator (Site not responding. Last check: 2007-10-07)
		In this article we discussthe term Hermitian as used in operator and matrix theory to refer to a certain kind of operator (or matrix).
		Matrix theorists sometimes refer to real Hermitian matrices as symmetricmatrices, since indeed they are symmetric with respect to the diagonal.
		In fact, Hermitian operators need not have any eigenvalues or eigenvectors atall.
	www.therfcc.org /hermitian-operator-155159.html (523 words)

	Encyclopedia article on Quantum mechanics [EncycloZine] (Site not responding. Last check: 2007-10-07)
		Each observable is represented by a densely-defined Hermitian (or self-adjoint) linear operator acting on the state space.
		During a measurement, the probability that a system collapses from a given initial state to a particular eigenstate is given by the square of the absolute value of the probability amplitudes between the initial and final states.
		They state that the state space of a system is a Hilbert space and the observables are Hermitian operators acting on that space, but do not tell us which Hilbert space or which operators.
	encyclozine.com /Quantum_physics (3734 words)

	Hermitian - The Jiggies Reference Guide (Site not responding. Last check: 2007-10-07)
		A continuous linear operator A: H → H on a Hilbert space A is called Hermitian or self-adjoint if
		For any two Hermitian operators A: H → H and B: H → H, and any element x of H holds the Cauchy-Bunyakovski-Schwarz inequality
		For this reason, one considers even more general Hermitian operators in the mathematical formulation of quantum mechanics: these operators are only defined on a dense subspace of a Hilbert space and don't have to be continuous.
	www.jiggies.com /reference/Hermitian (521 words)

	LMS Proceedings Abstract, paper PLMS 1401 (Site not responding. Last check: 2007-10-07)
		Define ${\rm HCl}(\mathcal{O} G)$ to be the Hermitian class group of $\mathcal{O} G$.
		When $d = 1$ and $\mathcal{O}$ is either $\mathbb{Z}$, $\mathbb{Z}_p$ or $\mathbb{R}$, we determine the image of $\chi_H(\mathcal{X}, G)$ in the adelic Hermitian classgroup ${\rm Ad\,HCl}(\mathbb{Z} G)$ by means of $\epsilon$-constants.
		We also show that in this case, the image in ${\rm Ad\,HCl}(\mathbb{Z} G)$ of a closely related Hermitian Euler characteristic $\chi_{H}(\mathcal{X}, G)(0)$ both determines and is determined by the $\epsilon_0$-constants of the symplectic representations of $G$.
	www.lms.ac.uk /publications/proceedings/abstracts/p1401a.html (187 words)

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