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Topic: Hermitian conjugate

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	Conjugate transpose - Wikipedia, the free encyclopedia
		It is often useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of complex conjugation.
		Another generalization is available: suppose A is a linear map from a complex vector space V to another W, then the complex conjugate linear map as well as the transposed linear map are defined, and we may thus take the conjugate transpose of A to be the complex conjugate of the transpose of A.
		It maps the conjugate dual of W to the conjugate dual of V.
	en.wikipedia.org /wiki/Conjugate_transpose (399 words)

	PlanetMath: conjugate transpose
		It is clear that for real matrices, the conjugate transpose coincides with the transpose.
		is also used for the conjugate transpose [2].
		This is version 5 of conjugate transpose, born on 2003-06-21, modified 2003-12-20.
	planetmath.org /encyclopedia/ConjugateTranspose.html (110 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)

	Conjugate transpose -- Facts, Info, and Encyclopedia article (Site not responding. Last check: 2007-10-20)
		It is often useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of (Click link for more info and facts about complex conjugation) complex conjugation.
		are both Hermitian and in fact (Click link for more info and facts about positive semi-definite) positive semi-definite.
		It maps the conjugate (Click link for more info and facts about dual) dual of W to the conjugate dual of V.
	www.absoluteastronomy.com /encyclopedia/c/co/conjugate_transpose.htm (499 words)

	CPC Licence Alert (Site not responding. Last check: 2007-10-20)
		The eigenproblem Hx = eSx where H and S are complex hermitian matrices with S positive definite has arisen in studies of the vibrational spectra of crystals and the molecular electronic structure of polymers.
		The metric S is decomposed by a complex variant of the Choleski decomposition S = LL+, where L+ denotes the hermitian conjugate of L. L is a complex lower triangular matrix with real diagonal elements.
		G is hermitian and its eigensolutions are obtained by Mueller's algorithm.
	www.cpc.cs.qub.ac.uk /summaries/AAAF.html (330 words)

	Hermitian and Skew Hermitian
		The conjugate of a complex matrix is the conjugate of all its entries.
		Since m is hermitian, replace the conjugate of ym with mz, where z is the conjugate of y.
		A skew hermitian operator is linear, and the set of skew hermitian operators on s forms a vector space.
	www.mathreference.com /la-sim,herm.html (793 words)

	Hermitian Operators (Site not responding. Last check: 2007-10-20)
		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)

	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)

	Hermitian adjoint (Site not responding. Last check: 2007-10-20)
		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 thedevelopement of the system with time.
		The concept is generalized from matrices of (real and/or complex) numbers to linear operators generally, and from vectorspaces to function spaces.
	www.therfcc.org /hermitian-adjoint-278821.html (328 words)

	Quaternion Quantum Mechanics
		Some of his terminology is a little different from mine. For example when his term quaternion corresponds to Hermitian quaternion.
		The advantage of the last form is that it avoids the conjugate of the wave function.
		It doubles the number of equations and might be explored as a generalization of the Dirac equation. Eq (37) and 33 (33) are special solutions.
	home.pcisys.net /~bestwork.1/QRW/QuaternionQuantumMechanics.htm (883 words)

	Maths -Conjugation - Martin Baker (Site not responding. Last check: 2007-10-20)
		The term 'conjugation' occurs in several fields such as grammar and biology, but in mathematics the term can be used in two ways:
		Conjugation of a complex number or quaternion or matrix is an operation which negates (multiplies by -1) the imaginary part.
		The adjoint matrix (sometimes also called the adjugate matrix, Hermitian transpose or conjugate transpose) is formed by taking the conjugate of each element in the matrix and then transposing the matrix.
	www.martinb.com /maths/algebra/principles/conjugation/index.htm (280 words)

	Matrices (Site not responding. Last check: 2007-10-20)
		All elements of a conjugate matrix are the complex conjugate of the original matrix. The conjugate matrix is commonly denoted by a superscript “*” (asterisk).
		The Hermitian matrix is the conjugate transpose of the original matrix. The Hermitian is commonly denoted with a superscript “H”. Note that the diagonal elements of a Hermitian matrix must be real.
		The identity matrix is a square (diagonal) matrix with all diagonal elements equal to 1 and all non-diagonal elements equal to 0. It is also called the unity matrix. The identity matrix is commonly denoted by
	www.electromagnetics.biz /Matrices.htm (691 words)

	sciforums.com - Hermitian conjugates, please help.
		This isn't even eligible to be Hermitian (it's missing a factor of i), so we expect it to be anti-Hermitian.
		You haven't made a mistake, but the mapping you have defined here Q# is not a linear operator, and therefore can't possible be the Hermitian adjoint (notice that 0#≠0).
		This defines a mapping Q# which depends on Q, but this mapping is not the Hermitian adjoint (which doesn't exist on this domain).
	www.sciforums.com /showthread.php?t=44017 (693 words)

	Re: hermitian operators for bears of little brain
		Charles Francis writes > > >HERMITIAN CONJUGATE >We have already seen that corresponding to each ket there is a bra, >known as the dual.
		Likewise corresponding to each operator there is a >Hermitian conjugate.
		What I would like to say is that is the conjugate of , unfortunately I can't see how to show this with confidence.
	www.lns.cornell.edu /spr/2003-06/msg0051792.html (629 words)

	hermitian operators for bears of little brain
		HERMITIAN CONJUGATE We have already seen that corresponding to each ket there is a bra, known as the dual.
		The action of the hermitian conjugate on the bra corresponds exactly to the action of the operator on the ket.
		Torture session: Show that is the complex conjugate of Definition As we are going to need it later, now seems a good time to mention that A is called a hermitian operator if A=A+.
	www.lns.cornell.edu /spr/2003-05/msg0051550.html (344 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)

	physics - Quantum mechanics
		Each observable is represented by a densely-defined Hermitian (or self-adjoint) linear operator acting on the state space.
		Each eigenstate of an observable corresponds to an eigenvector of the operator, and the associated eigenvalue corresponds to the value of the observable in that eigenstate.
		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.
	www.physicsdaily.com /physics/Quantum_mechanics (3978 words)

	[No title] (Site not responding. Last check: 2007-10-20)
		Date: Wed, 26 Aug 1998 17:07:10 -0400 (EDT) The reason MATLAB returns the conjugate transpose (Hermitian conjugate) for complex matrices is that this is in fact the propper generalization of tranposition when the underlying field is the complex numbers.
		When that underlying field degenerates to the reals, the Hermitian conjugate becomes the transpose (over the reals) that we all know and love.
		So it is very rare in the signals and systems arena to encounter the transpose of a complex matrix.
	www.octave.org /mailing-lists/bug-octave/1998/325 (213 words)

	Bra-ket notation
		is called the hermitian conjugate of operator A.
		Hermitian operators play a central role in quantum theory.
		Consider a hermitian operator X, whose eigenstates
	www.physics.unlv.edu /~bernard/phy721_99/tex_notes/node6.html (423 words)

	Re: hermitian operators for bears of little brain
		In message , Oz writes: >Charles Francis writes: >>HERMITIAN CONJUGATE >>We have already seen that corresponding to each ket there is a bra, >>known as the dual.
		That's because I don't know what it is! Complex conjugation is something you can apply to a number, and it gives you another number.
		This is the definition of a hermitian matrix, the torture session applied to any matrix.
	www.lns.cornell.edu /spr/2003-07/msg0052633.html (511 words)

	Read about Conjugate transpose at WorldVillage Encyclopedia. Research Conjugate transpose and learn about Conjugate ... (Site not responding. Last check: 2007-10-20)
		It is often useful to think of square complex matrices as "generalized complex numbers", and of the conjugate transpose as a generalization of
		complex conjugate linear map as well as the
		transposed linear map are defined, and we may thus take the conjugate transpose of A to be the complex conjugate of the transpose of A.
	encyclopedia.worldvillage.com /s/b/Conjugate_transpose (320 words)

	ipedia.com: Conjugate transpose Article (Site not responding. Last check: 2007-10-20)
		In mathematics, the conjugate transpose or adjoint of an m -by- n matrix A with complex entries is the n -by- m matrix A * obtained from A by taking the transpose and then taking the complex conjugate...
		When working in Hilbert space, especially with the bra-ket notation, the adjoint operator - called the Hermitian Conjugate, denoted as, is defined by the relation
		The term Hermitian conjugate transpose is used since if than A is called an Hermitian operator.
	www.ipedia.com /conjugate_transpose.html (427 words)

	Re: hermitian operators for bears of little brain
		In message <6WBCa.30080$Io.2549280@newsread2.prod.itd.earthlink.net>, Mikhail Erementchouk writes >"Charles Francis" wrote >> HERMITIAN CONJUGATE > >> We have already seen that corresponding to each ket there is a bra, >> known as the dual.
		Likewise corresponding to each operator there is a >> Hermitian conjugate.
		The action of the hermitian conjugate on the bra >> corresponds exactly to the action of the operator on the ket.
	www.lns.cornell.edu /spr/2003-06/msg0051798.html (366 words)

	Conjugate transpose (Site not responding. Last check: 2007-10-20)
		When working in Hilbert space, especially with the bra-ket notation, the adjoint operator - called the Hermitian Conjugate, denoted as
		The term Hermitian conjugate transpose is used since if
		It has also been stated in which way the above can be related to the notion of a pair of adjoint functors in category theory.
	www.sciencedaily.com /encyclopedia/conjugate_transpose (415 words)

	Hermitian Conjugation
		Thus the transpose of an antilinear map whose outputs are linear will be a linear map whose outputs are antilinear.
		Thus the Hermitian transpose is, for antilinear maps (between linear spaces), what (ordinary) transpose is for linear maps.
		None the less, it is usual to work with symmetric inner products; albeit now Hermitian symmetric, rather than transpose symmetric.
	www.chaos.org.uk /~eddy/math/linear/Hermite.html (2542 words)

	Hermitian adjoint
		For every linear operator (which is bounded, see operator's norm), we define the Hermitian adjoint, which we mark by (pronounced "A dagger") by
		The last formula is very useful for computing a norm of operator since is self-adjoint (Hermitian).
		The Hamiltonian (energy) of physical systems is the most important Hermitian operator, since its govern the developement of the system with time.
	www.faqfolio.com /faqfolio/h/he/hermitian_adjoint.html (346 words)

	Notes 6
		When we were dealing with operators, I had a rather different looking definition of the adjoint (also called the Hermitian conjugate) than the matrix one.
		Since we just showed p^2 is Hermitian, and in an earlier example we showed that V(x) is Hermitian, it means that the Hamiltonian
		Proof of theorem 4) The eigenvectors of Hermitian operators belonging to distinct eigenvalues are orthogonal.
	www.colorado.edu /physics/phys3220/3220_fa97/notes/notes6/3220_notes6_4.html (632 words)

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