SMART Information Retrieval System - Revision history

en>Thyriakus at 21:55, 24 July 2014

2014-07-24T21:55:37Z

← Older revision		Revision as of 23:55, 24 July 2014
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	In [[linear algebra]], a [[square matrix]] with [[Complex number\|complex]] entries is said to be '''skew-Hermitian''' or '''antihermitian''' if its [[conjugate transpose]] is equal to its negative.<ref>{{harvtxt\|Horn\|Johnson\|1985}}, §4.1.1; {{harvtxt\|Meyer\|2000}}, §3.2</ref> That is, the ~~matrix~~ '~~'A'' is skew-Hermitian if it satisfies the relation~~		These author is known by the name of Gabrielle Lattimer though she doesn't actually like being called in this way. For years she's been working even though a library assistant. To bake is something that she's been doing for yrs. For years she's been tough to adapt in Massachusetts. She is running and examining a blog here: http://circuspartypanama.com<br><br>Here is my web-site [http://circuspartypanama.com Clash Of Clans Hack Download]

	~~:<math>A^\dagger = -A,\;</math>~~

	~~where <math>\dagger</math> denotes the conjugate transpose of a matrix. In component form,~~ this ~~means that~~
	~~:<math>a_{i,j} = -\overline{a_{j,i}},</math>~~
	~~for all ''i'' and ''j'', where ''a''<sub>''i'',''j''</sub> is the ''i'',''j''-th entry of ''A'', and the overline denotes [[complex conjugate\|complex conjugation]]~~.

	Skew-Hermitian matrices can be understood as the complex versions of real [[Skew-symmetric matrix\|skew-symmetric matrices]], or as the matrix analogue of the purely imaginary numbers.<ref name=HJ85S412>{{harvtxt\|Horn\|Johnson\|1985}}, §4.1.2</ref> All skew-Hermitian <var>n</var>×<var>n</var> matrices form the '~~''u'''(<var>n</var>) [[Lie algebra]], which corresponds to the Lie group [[Unitary group\|U(<var>n</var>)]].~~
	~~The concept can be generalized to include [[linear transformation]]~~s ~~of any [[complex number \| complex]] [[vector space]] with~~ a ~~[[sesquilinear]] [[Norm (mathematics)\|norm]]~~.

	~~== Example ==~~

	~~For example, the following matrix~~ is ~~skew-Hermitian:~~
	~~:<math>\begin{bmatrix} -i & 2 + i \\ -(2 - i) & 0 \end{bmatrix}</math>~~

	~~== Properties ==~~

	* The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero. Furthermore, skew-Hermitian matrices are [[normal matrix\|normal]]. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.<ref>{{harvtxt\|Horn\|Johnson\|1985}}, §2.5.2, §2.5.4</ref>
	* All entries on the [[main diagonal]] of a skew-Hermitian matrix have to be pure [[imaginary number\|imaginary]], i.e., on the imaginary axis (the number zero is also considered purely imaginary).<ref>{{harvtxt\|Meyer\|2000}}, Exercise 3.2.5</ref>
	* If '~~'A, B'' are skew-Hermitian, then ''aA + bB'' is skew-Hermitian~~ for ~~all [[real number\|real]] [[scalar (mathematics)\|scalars]] ''a'' and ''b''~~.~~<ref name=HJ85S411>{{harvtxt\|Horn\|Johnson\|1985}}, §4.1.1</ref>~~
	* If '~~'A'' is skew-Hermitian, then both ''i A'' and −''i A'' are [[Hermitian matrix\|Hermitian]]~~.~~<ref name=HJ85S411/>~~
	* If ''A'' is skew-Hermitian, then ''A''<sup>''k''</sup> is Hermitian if ''k'' is an even integer and skew-Hermitian if ''k'' is ~~an odd integer.~~
	* An arbitrary (square) matrix ''C'' can uniquely be written as the sum of a Hermitian matrix ''A'' and a ~~skew-Hermitian matrix ''B''~~:~~<ref name=HJ85S412~~/>
	~~::<math>C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^\dagger) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^\dagger)~~.<~~/math~~>
	* If ''A'' is skew-Hermitian, then e<~~sup>''A''</sup~~> is ~~[[unitary matrix\|unitary]].~~
	* The space of skew-~~Hermitian matrices forms the~~ [~~[Lie algebra]] u(''n'') of the [[Lie group]] U(''n'').~~

	~~==See also==~~
	*[[Bivector (complex)]]
	*[[Hermitian matrix]]
	*[[Normal matrix]]
	*[[Skew-symmetric matrix]]
	*[[Unitary matrix]]

	~~==Notes==~~
	~~<references/>~~

	~~==References==~~
	* {{Citation \| last1=Horn \| first1=Roger A. \| last2=Johnson \| first2=Charles R. \| title=Matrix Analysis \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-38632-6 \| year=1985}}.
	* {{Citation \| last1=Meyer \| first1=Carl D. \| title=Matrix Analysis and Applied Linear Algebra \| url=http://~~www.matrixanalysis~~.com~~/ \| publisher=[[Society for Industrial and Applied Mathematics\|SIAM]] \| isbn=978-0-89871-454-8 \| year=2000}}.~~

	~~[[Category:Matrices]~~]

en>BG19bot: WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB

2013-03-10T02:00:14Z

WP:CHECKWIKI error fix for #61. Punctuation goes before References. Do general fixes if a problem exists. - using AWB

← Older revision		Revision as of 04:00, 10 March 2013
Line 1:		Line 1:
	~~This can~~ a ~~strategy and as well as battle activation where your entire family must manage your~~ be ~~the owner of tribe and also prevent leakages. You have~~ to ~~build constructions which will definately provide protection for your soldiers along with our instruction. First think straight on your protection and simply after~~ its ~~recently been taken treatment~~. ~~You will need to move forward offering the criminal offense process~~. ~~As well as your company Military facilities, you in addition need to keep in minds the way your indigneous group is certainly going~~. ~~For instance~~, ~~collecting time and energy as well as raising your own tribe may be the key to good improvements~~.<~~br><br~~>		In [[linear algebra]], a [[square matrix]] with [[Complex number\|complex]] entries is said to be '''skew-Hermitian''' or '''antihermitian''' if its [[conjugate transpose]] is equal to its negative.<ref>{{harvtxt\|Horn\|Johnson\|1985}}, §4.1.1; {{harvtxt\|Meyer\|2000}}, §3.2</ref> That is, the matrix ''A'' is skew-Hermitian if it satisfies the relation

	~~Host~~ a ~~video game champion~~. ~~These can be a lot amongst fun~~ for ~~you together gaming friends. You may do this online~~, ~~at your house . or at~~ a ~~Buddy~~'~~s place~~. ~~Serve amazing snacks and get~~ the ~~number~~ of ~~people~~ as ~~you can possibly involved~~. ~~This makes agreat way to enjoy ones game playing with friends~~.<br><br>~~If you happen to getting a online mission for your little one~~, ~~look for one exactly~~ which ~~enables numerous customers~~ to ~~do with each other~~. ~~Video gaming~~ can be a ~~solitary action~~. ~~Nevertheless~~, it is ~~important for you to motivate your youngster growing to be social~~, ~~and multi~~-~~player clash of clans hack is capable to do that~~. ~~They enable sisters~~ and ~~brothers while buddies to all among take a moment and laugh and compete jointly~~.<br><br>~~In cases where you're playing~~ a ~~gameplay online, and you execute across another player what kind of person seems~~ to be ~~disheartening other players~~ (~~or you, in particular~~) ~~intentionally~~, ~~never will take it personally~~. ~~This is called "Griefing~~,~~" and it~~'~~s the is a little bit equivalent of Internet trolling. Griefers~~ are ~~mearly out for negative attention~~, ~~and you give the entire group what they~~'~~re looking designed~~ for ~~if you interact~~ all ~~of them. Don~~'~~t get emotionally utilized in what~~'~~s happening in addition to simply try to ignore it~~.<br><br>If ~~you~~ are ~~looking Conflict of Families Jewels Free~~, ~~or you~~'~~re just buying a Compromise Conflict of Tribes, currently has the smartest choice while on the internet, absolutely free as well as only takes a few minutes to get all all these.~~<br><br>It'~~s to select the required xbox game gaming function~~. In the ~~beginning, you should think about your standard requirements like~~ a ~~video game player, following check out the additional features made available from together unit you are deciding on. Consider investigating on~~-~~line. Check consumer reviews to ascertain if former gamers have [http~~:/~~/browse.Deviantart.com/?q~~=~~discovered~~+~~difficulties discovered difficulties]~~ with ~~the unit~~. ~~Prior to buying a game process~~, ~~you should know to the extent that you are able with regarding it.~~<br><br>~~While your village grows, possess to to explore uncharted areas for Gold and Sound which are~~ the ~~8 key resources you may want to expect to require here in start~~ of the ~~hobby~~ ( ~~addititionally there is Stone resource, that your business discover later inside the type of game~~ ). ~~Into your exploration, you will be able to expect to stumble upon many islands whereby that~~ [http://www.~~thefreedictionary~~.com/~~villages+happen villages happen~~] ~~to stay held captive under BlackGuard slavery and you profit from free Gold possessions if they are unoccupied If you have any inquiries pertaining to where and how to use clash of clans hack tool (~~[~~http~~:~~//prometeu.net the full details~~]~~), you can get in touch with us at our web site. .~~		:<math>A^\dagger = -A,\;</math>

			where <math>\dagger</math> denotes the conjugate transpose of a matrix. In component form, this means that
			:<math>a_{i,j} = -\overline{a_{j,i}},</math>
			for all ''i'' and ''j'', where ''a''<sub>''i'',''j''</sub> is the ''i'',''j''-th entry of ''A'', and the overline denotes [[complex conjugate\|complex conjugation]].

			Skew-Hermitian matrices can be understood as the complex versions of real [[Skew-symmetric matrix\|skew-symmetric matrices]], or as the matrix analogue of the purely imaginary numbers.<ref name=HJ85S412>{{harvtxt\|Horn\|Johnson\|1985}}, §4.1.2</ref> All skew-Hermitian <var>n</var>×<var>n</var> matrices form the '''u'''(<var>n</var>) [[Lie algebra]], which corresponds to the Lie group [[Unitary group\|U(<var>n</var>)]].
			The concept can be generalized to include [[linear transformation]]s of any [[complex number \| complex]] [[vector space]] with a [[sesquilinear]] [[Norm (mathematics)\|norm]].

			== Example ==

			For example, the following matrix is skew-Hermitian:
			:<math>\begin{bmatrix} -i & 2 + i \\ -(2 - i) & 0 \end{bmatrix}</math>

			== Properties ==

			* The eigenvalues of a skew-Hermitian matrix are all purely imaginary or zero. Furthermore, skew-Hermitian matrices are [[normal matrix\|normal]]. Hence they are diagonalizable and their eigenvectors for distinct eigenvalues must be orthogonal.<ref>{{harvtxt\|Horn\|Johnson\|1985}}, §2.5.2, §2.5.4</ref>
			* All entries on the [[main diagonal]] of a skew-Hermitian matrix have to be pure [[imaginary number\|imaginary]], i.e., on the imaginary axis (the number zero is also considered purely imaginary).<ref>{{harvtxt\|Meyer\|2000}}, Exercise 3.2.5</ref>
			* If ''A, B'' are skew-Hermitian, then ''aA + bB'' is skew-Hermitian for all [[real number\|real]] [[scalar (mathematics)\|scalars]] ''a'' and ''b''.<ref name=HJ85S411>{{harvtxt\|Horn\|Johnson\|1985}}, §4.1.1</ref>
			* If ''A'' is skew-Hermitian, then both ''i A'' and −''i A'' are [[Hermitian matrix\|Hermitian]].<ref name=HJ85S411/>
			* If ''A'' is skew-Hermitian, then ''A''<sup>''k''</sup> is Hermitian if ''k'' is an even integer and skew-Hermitian if ''k'' is an odd integer.
			* An arbitrary (square) matrix ''C'' can uniquely be written as the sum of a Hermitian matrix ''A'' and a skew-Hermitian matrix ''B'':<ref name=HJ85S412/>
			::<math>C = A+B \quad\mbox{with}\quad A = \frac{1}{2}(C + C^\dagger) \quad\mbox{and}\quad B = \frac{1}{2}(C - C^\dagger).</math>
			* If ''A'' is skew-Hermitian, then e<sup>''A''</sup> is [[unitary matrix\|unitary]].
			* The space of skew-Hermitian matrices forms the [[Lie algebra]] u(''n'') of the [[Lie group]] U(''n'').

			==See also==
			*[[Bivector (complex)]]
			*[[Hermitian matrix]]
			*[[Normal matrix]]
			*[[Skew-symmetric matrix]]
			*[[Unitary matrix]]

			==Notes==
			<references/>

			==References==
			* {{Citation \| last1=Horn \| first1=Roger A. \| last2=Johnson \| first2=Charles R. \| title=Matrix Analysis \| publisher=[[Cambridge University Press]] \| isbn=978-0-521-38632-6 \| year=1985}}.
			* {{Citation \| last1=Meyer \| first1=Carl D. \| title=Matrix Analysis and Applied Linear Algebra \| url=http://www.matrixanalysis.com/ \| publisher=[[Society for Industrial and Applied Mathematics\|SIAM]] \| isbn=978-0-89871-454-8 \| year=2000}}.

			[[Category:Matrices]]

en>Thorwald: "External links" section is always last + fmt

2012-05-23T19:16:57Z

"External links" section is always last + fmt

New page

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