Detrended fluctuation analysis - Revision history

213.105.234.47 at 14:00, 12 January 2015

2015-01-12T14:00:21Z

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178.38.195.237: /* Calculation */

2014-03-04T14:03:07Z

Calculation

← Older revision		Revision as of 16:03, 4 March 2014
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	~~In [[mathematics]],~~ '''~~persymmetric matrix''' may refer to:~~		The author's name is Andera and she thinks it seems fairly good. I've usually cherished residing in Alaska. Since I was 18 I've been working as a bookkeeper but soon my wife and I will start our own business. One of the very best things in the globe for him is doing ballet and he'll be starting something else alongside with it.<br><br>Here is [http://1.234.36.240/fxac/m001_2/7330 tarot card readings] my homepage [http://myfusionprofits.com/groups/find-out-about-self-improvement-tips-here/ free tarot readings] - [http://findyourflirt.net/index.php?m=member_profile&p=profile&id=117823 http://findyourflirt.net/],
	~~# a [[square matrix]] which is symmetric in the northeast-to-southwest diagonal; or~~
	~~# a square matrix such that the values on each line perpendicular to the main diagonal are the same for~~ a ~~given line~~.
	~~The first definition is~~ the ~~most common~~ in the ~~recent literature. The designation "[[Hankel matrix]]"~~ is ~~often used for matrices satisfying the property in the second definition~~.

	~~== Definition 1 ==~~
	~~Let ''A'' = (''a''~~<~~sub~~>~~''i''⁣''j''~~<~~/sub~~>~~) be an ''n'' × ''n'' matrix.~~ ~~The first definition of ''persymmetric'' requires that~~
	:~~<math> a_{ij} = a_{n-j+1,n-i+~~1~~} </math> for all ''i'', ''j''~~.~~<ref>{{citation \| first1=Gene H~~. ~~\| last1=Golub \| author1-link=Gene H~~. ~~Golub \| first2=Charles F~~. ~~\| last2=Van Loan \| author2~~-~~link=Charles F. Van Loan \| year=1996 \| title=Matrix Computations \| edition=3rd \| publisher=Johns Hopkins \| place=Baltimore \| isbn=978~~-0-~~8018~~-~~5414~~-~~9}}. See page 193.<~~/~~ref>~~
	~~For example, 5~~-~~by-5 persymmetric matrices are of the form~~
	:~~<math> A = \begin{bmatrix}~~
	~~a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\~~
	~~a_{21} & a_{22} & a_{23} & a_{24} & a_{14} \\~~
	~~a_{31} & a_{32} & a_{33} & a_{23} & a_{13} \\~~
	~~a_{41} & a_{42} & a_{32} & a_{22} & a_{12} \\~~
	~~a_{51} & a_{41} & a_{31} & a_{21} & a_{11}~~
	~~\end{bmatrix}. <~~/~~math>~~

	~~This can be equivalently expressed as ''AJ = JA''<sup>T<~~/~~sup> where ''J'' is the [[exchange matrix]]~~.

	~~A [[symmetric matrix]] is a matrix whose values are symmetric in the northwest-to-southeast diagonal. If a symmetric matrix is rotated by 90°, it becomes a persymmetric matrix~~. ~~Symmetric persymmetric matrices are sometimes called [[bisymmetric matrix\|bisymmetric matrices]].~~

	== ~~Definition 2~~ ==
	~~{{details\|Hankel matrix}}~~

	The second definition is due to [[Thomas Muir (mathematician)\|Thomas Muir]].<ref name="muir">{{Citation\|last=Muir\|first=Thomas\|title=Treatise on the Theory of Determinants\|page= 419\|publisher= Dover Press\|year= 1960}}</~~ref> It says that the square matrix ''A'' = (''a''<sub>''ij''<~~/~~sub>) is persymmetric if ''a''<sub>''ij''</sub> depends only on ''i'' + ''j''. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form~~
	~~:<math> A = \begin{bmatrix}~~
	~~r_1 & r_2 & r_3 & \cdots & r_n \\~~
	~~r_2 & r_3 & r_4 & \cdots & r_{n+1} \\~~
	~~r_3 & r_4 & r_5 & \cdots & r_{n+2} \\~~
	~~\vdots & \vdots & \vdots & \ddots & \vdots \\~~
	~~r_n & r_{n+1} & r_{n+2} & \cdots & r_{2n-1}~~
	~~\end{bmatrix}~~.
	</~~math>~~
	~~A '''persymmetric determinant''' is the [[determinant~~]~~] of a persymmetric matrix.<ref name="muir"/>~~

	~~A matrix for which the values on each line parallel to the main diagonal are constant~~, ~~is called a [[Toeplitz matrix]].~~

	~~==References==~~
	~~{{Reflist}}~~

	~~[[Category:Determinants]]~~
	~~[[Category:Matrices]]~~

en>Anrnusna at 22:42, 23 September 2013

2013-09-23T22:42:48Z

← Older revision		Revision as of 00:42, 24 September 2013
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	~~Andrew Simcox~~ is the ~~title his parents gave him and he totally enjoys this title~~. ~~To play domino~~ is ~~something I really enjoy doing~~. ~~Invoicing~~ is ~~what I do~~ for a ~~living but I~~'~~ve always wanted my personal company~~. ~~Mississippi~~ is the ~~only place I've been residing~~ in ~~but I will have~~ to ~~transfer in~~ a ~~yr or two~~.<br><br>~~My blog post: psychic chat online~~ (~~[http:~~//~~appin~~.co.kr/~~board_Zqtv22~~/~~688025 visit~~ the ~~up coming document~~])		In [[mathematics]], '''persymmetric matrix''' may refer to:
			# a [[square matrix]] which is symmetric in the northeast-to-southwest diagonal; or
			# a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.
			The first definition is the most common in the recent literature. The designation "[[Hankel matrix]]" is often used for matrices satisfying the property in the second definition.

			== Definition 1 ==
			Let ''A'' = (''a''<sub>''i''⁣''j''</sub>) be an ''n'' × ''n'' matrix. The first definition of ''persymmetric'' requires that
			:<math> a_{ij} = a_{n-j+1,n-i+1} </math> for all ''i'', ''j''.<ref>{{citation \| first1=Gene H. \| last1=Golub \| author1-link=Gene H. Golub \| first2=Charles F. \| last2=Van Loan \| author2-link=Charles F. Van Loan \| year=1996 \| title=Matrix Computations \| edition=3rd \| publisher=Johns Hopkins \| place=Baltimore \| isbn=978-0-8018-5414-9}}. See page 193.</ref>
			For example, 5-by-5 persymmetric matrices are of the form
			:<math> A = \begin{bmatrix}
			a_{11} & a_{12} & a_{13} & a_{14} & a_{15} \\
			a_{21} & a_{22} & a_{23} & a_{24} & a_{14} \\
			a_{31} & a_{32} & a_{33} & a_{23} & a_{13} \\
			a_{41} & a_{42} & a_{32} & a_{22} & a_{12} \\
			a_{51} & a_{41} & a_{31} & a_{21} & a_{11}
			\end{bmatrix}. </math>

			This can be equivalently expressed as ''AJ = JA''<sup>T</sup> where ''J'' is the [[exchange matrix]].

			A [[symmetric matrix]] is a matrix whose values are symmetric in the northwest-to-southeast diagonal. If a symmetric matrix is rotated by 90°, it becomes a persymmetric matrix. Symmetric persymmetric matrices are sometimes called [[bisymmetric matrix\|bisymmetric matrices]].

			== Definition 2 ==
			{{details\|Hankel matrix}}

			The second definition is due to [[Thomas Muir (mathematician)\|Thomas Muir]].<ref name="muir">{{Citation\|last=Muir\|first=Thomas\|title=Treatise on the Theory of Determinants\|page= 419\|publisher= Dover Press\|year= 1960}}</ref> It says that the square matrix ''A'' = (''a''<sub>''ij''</sub>) is persymmetric if ''a''<sub>''ij''</sub> depends only on ''i'' + ''j''. Persymmetric matrices in this sense, or Hankel matrices as they are often called, are of the form
			:<math> A = \begin{bmatrix}
			r_1 & r_2 & r_3 & \cdots & r_n \\
			r_2 & r_3 & r_4 & \cdots & r_{n+1} \\
			r_3 & r_4 & r_5 & \cdots & r_{n+2} \\
			\vdots & \vdots & \vdots & \ddots & \vdots \\
			r_n & r_{n+1} & r_{n+2} & \cdots & r_{2n-1}
			\end{bmatrix}.
			</math>
			A '''persymmetric determinant''' is the [[determinant]] of a persymmetric matrix.<ref name="muir"/>

			A matrix for which the values on each line parallel to the main diagonal are constant, is called a [[Toeplitz matrix]].

			==References==
			{{Reflist}}

			[[Category:Determinants]]
			[[Category:Matrices]]

en>FrescoBot: Bot: fixing section wikilinks and minor changes

2012-02-25T09:35:20Z

Bot: fixing section wikilinks and minor changes

New page

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