Unduloid - Revision history

en>AnomieBOT: Dating maintenance tags: {{Unreferenced}}

2014-10-07T19:13:57Z

Dating maintenance tags: {{Unreferenced}}

← Older revision		Revision as of 21:13, 7 October 2014
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	~~In [[linear algebra]], skew-Hamiltonian matrices are special [[Matrix (mathematics)\|matrices]] which correspond to [[skew-symmetric]] [[bilinear form]]~~s ~~on a [[symplectic vector space]].~~		The author's name is Andera and she thinks it sounds fairly great. What me and my family [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ love psychics] is bungee leaping but I've been taking on new issues recently. North Carolina is the location he loves most but now he is considering other choices. He is an info officer.

	~~Let ''V'' be a [[vector space]], equipped with a [[Symplectic vector space\|symplectic form]] <math>\Omega</math>. Such a space must be even-dimensional. A linear map <math>A:\; V \mapsto V</math>~~ is ~~called '''a skew-Hamiltonian operator''' with respect to <math>\Omega</math> if the form <math>x, y \mapsto \Omega(A(x), y)</math> is skew-symmetric.~~

	~~Choose a basis <math> e_1, ... e_{2n}</math> in ''V'', such that <math>\Omega</math> is written as <math>\sum_i e_i \wedge e_{n+i}</math>~~. ~~Then a linear operator is skew-Hamiltonian with respect to <math>\Omega</math> if and only if its matrix ''A'' satisfies <math>A^T J = J A</math>, where ''J'' is the skew-symmetric matrix~~

	~~:<math>J=~~
	~~\begin{bmatrix}~~
	~~0 & I_n \\~~
	~~-I_n & 0 \\~~
	~~\end{bmatrix}</math>~~

	and ~~''I<sub>n</sub>'' is the <math>n\times n</math> [[identity matrix]].<ref name=waterhouse>[[William C. Waterhouse]],~~ [http://~~linkinghub.elsevier~~.com/~~retrieve~~/~~pii/S0024379504004410 ''The structure of alternating~~-~~Hamiltonian matrices''], Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385~~-~~390<~~/~~ref> Such matrices are called '''skew-Hamiltonian'''.~~

	~~The square of a [[Hamiltonian matrix]~~] is ~~skew-Hamiltonian~~. ~~The converse~~ is ~~also true: every skew-Hamiltonian matrix can be obtained as~~ the ~~square of a Hamiltonian matrix~~.~~<ref name=waterhouse/><ref>~~
	~~Heike Faßbender, D. Steven Mackey, Niloufer Mackey and Hongguo Xu~~
	~~[http://www.icm.tu-bs.de/~hfassben/papers/hamsqrt~~.~~pdf Hamiltonian Square Roots of Skew-Hamiltonian Matrices,]~~
	~~Linear Algebra and its Applications 287, pp. 125 - 159, 1999</ref>~~

	~~==Notes==~~

	~~<references />~~

	~~[[Category:Matrices]]~~
	~~[[Category:Linear algebra]]~~


	~~{{Linear-algebra-stub}}~~

en>Addbot: Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3505271

2013-03-17T01:35:08Z

Bot: Migrating 2 interwiki links, now provided by Wikidata on d:q3505271

← Older revision		Revision as of 03:35, 17 March 2013
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	~~Hello!~~ Let ~~me start by saying my name~~ - ~~Ron Stephenson~~. ~~Alabama~~ is ~~exactly where he and his wife reside and he has every thing~~ that ~~he requirements there~~. ~~Bottle tops gathering~~ is ~~the~~ only ~~pastime his wife doesn~~'~~t approve of. Managing people~~ is ~~what~~ I ~~do in my day occupation~~.<br><br>~~My weblog~~ ... [http://~~Www~~.~~jbcopy~~.~~com~~/~~node~~/~~430 extended auto warranty~~]		In [[linear algebra]], skew-Hamiltonian matrices are special [[Matrix (mathematics)\|matrices]] which correspond to [[skew-symmetric]] [[bilinear form]]s on a [[symplectic vector space]].

			Let ''V'' be a [[vector space]], equipped with a [[Symplectic vector space\|symplectic form]] <math>\Omega</math>. Such a space must be even-dimensional. A linear map <math>A:\; V \mapsto V</math> is called '''a skew-Hamiltonian operator''' with respect to <math>\Omega</math> if the form <math>x, y \mapsto \Omega(A(x), y)</math> is skew-symmetric.

			Choose a basis <math> e_1, ... e_{2n}</math> in ''V'', such that <math>\Omega</math> is written as <math>\sum_i e_i \wedge e_{n+i}</math>. Then a linear operator is skew-Hamiltonian with respect to <math>\Omega</math> if and only if its matrix ''A'' satisfies <math>A^T J = J A</math>, where ''J'' is the skew-symmetric matrix

			:<math>J=
			\begin{bmatrix}
			0 & I_n \\
			-I_n & 0 \\
			\end{bmatrix}</math>

			and ''I<sub>n</sub>'' is the <math>n\times n</math> [[identity matrix]].<ref name=waterhouse>[[William C. Waterhouse]], [http://linkinghub.elsevier.com/retrieve/pii/S0024379504004410 ''The structure of alternating-Hamiltonian matrices''], Linear Algebra and its Applications, Volume 396, 1 February 2005, Pages 385-390</ref> Such matrices are called '''skew-Hamiltonian'''.

			The square of a [[Hamiltonian matrix]] is skew-Hamiltonian. The converse is also true: every skew-Hamiltonian matrix can be obtained as the square of a Hamiltonian matrix.<ref name=waterhouse/><ref>
			Heike Faßbender, D. Steven Mackey, Niloufer Mackey and Hongguo Xu
			[http://www.icm.tu-bs.de/~hfassben/papers/hamsqrt.pdf Hamiltonian Square Roots of Skew-Hamiltonian Matrices,]
			Linear Algebra and its Applications 287, pp. 125 - 159, 1999</ref>

			==Notes==

			<references />

			[[Category:Matrices]]
			[[Category:Linear algebra]]


			{{Linear-algebra-stub}}

en>Luckas-bot: r2.7.1) (Robot: Adding sl:Onduloid

2011-12-28T20:46:14Z

r2.7.1) (Robot: Adding sl:Onduloid

New page

Hello! Let me start by saying my name - Ron Stephenson. Alabama is exactly where he and his wife reside and he has every thing that he requirements there. Bottle tops gathering is the only pastime his wife doesn't approve of. Managing people is what I do in my day occupation.<br><br>My weblog ... [http://Www.jbcopy.com/node/430 extended auto warranty]