Hubble volume - Revision history

en>Tetra quark: /* See also */ added the cosmology portal

2015-01-08T19:05:17Z

See also: added the cosmology portal

← Older revision		Revision as of 21:05, 8 January 2015
Line 1:		Line 1:
	~~In [[mathematics]], particularly [[Matrix (mathematics)\|matrix theory]], the ''n×n'' '''Lehmer matrix''' (named after [[Derrick Henry Lehmer]])~~ is ~~the constant [[symmetric matrix]] defined~~ by		Royal Votaw is my name but I by no means really favored that title. The job he's been occupying for many years is a messenger. My house is now in Kansas. The favorite pastime for him and his children is to generate and now he is trying to make money with it.<br><br>my blog [http://Family.Ec-Win.ir/index.php?do=/profile-2088/info/ ec-win.ir]
	~~:<math>A_{ij} =~~
	~~\begin{cases}~~
	~~i/j, & j\ge i \\~~
	~~j/i, & j<i~~.
	~~\end{cases}~~
	~~</math>~~

	~~Alternatively, this may be written as~~
	~~:<math>A_{ij} = \frac{\mbox{min}(i,j)}{\mbox{max}(i,j)}.</math>~~

	~~==Properties==~~

	~~As can be seen in the examples section, if ''A'' is an ''n×n'' Lehmer matrix and ''B'' is an ''m×m'' Lehmer matrix, then ''A'~~' is a ~~[[submatrix]] of ''B'' whenever ''m''>''n''~~. ~~The values of elements diminish toward zero away from the diagonal, where all elements have value 1.~~

	~~Interestingly, the [[matrix inverse\|inverse]] of a Lehmer matrix~~ is ~~a [[tridiagonal matrix]], where the [[superdiagonal]] and [[subdiagonal]] have strictly negative entries~~. ~~Consider again the ''n×n'' ''A''~~ and ~~''m×m'' ''B'' Lehmer matrices, where ''m''>''n''. A rather peculiar property of their inverses~~ is ~~that ''A<sup>-1</sup>'' is ''nearly'' a submatrix of ''B<sup>-1</sup>'', except for the ''A<sub>n,n</sub>'' element, which~~ is ~~not equal~~ to ~~''B~~<~~sub~~>~~m,m~~<~~/sub~~>~~''.~~

	~~A Lehmer matrix of order ''n'' has~~ [~~[trace of a matrix\|trace]] ''n''.~~

	~~==Examples==~~
	~~The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.~~
	:~~<math>~~
	~~\begin{array}{lllll}~~
	~~A_2=\begin{pmatrix}~~
	~~1 & 1~~/~~2 \\~~
	1/~~2 & 1~~
	~~\end{pmatrix};~~
	&
	~~A_2^{~~-~~1}=\begin{pmatrix}~~
	4/~~3 & -2/3 \\~~
	~~-2/3 & {\color{BrickRed}\mathbf{4/3}}~~
	~~\end{pmatrix};~~

	\\
	\\

	~~A_3~~=~~\begin{pmatrix}~~
	~~1 & 1~~/~~2 & 1/3 \\~~
	~~1/2 & 1 & 2/3 \\~~
	~~1/3 & 2/3 & 1~~
	~~\end{pmatrix};~~
	&
	~~A_3^{~~-~~1}=\begin{pmatrix}~~
	4/~~3 & -2~~/~~3 & \\~~
	-~~2/3 & 32/15 & -6/5 \\~~
	~~& -6/5 & {\color{BrickRed}\mathbf{9/5}}~~
	~~\end{pmatrix};~~

	\\
	\\

	~~A_4=\begin{pmatrix}~~
	~~1 & 1/2 & 1/3 & 1/4 \\~~
	~~1/2 & 1 & 2/3 & 1/2 \\~~
	~~1/3 & 2/3 & 1 & 3/4 \\~~
	~~1/4 & 1/2 & 3/4 & 1~~
	~~\end{pmatrix};~~
	&
	~~A_4^{-1}=\begin{pmatrix}~~
	~~4/3 & -2/3 & & \\~~
	~~-2/3 & 32/15 & -6/5 & \\~~
	~~& -6/5 & 108/35 & -12/7 \\~~
	~~& & -12/7 & {\color{BrickRed}\mathbf{16/7}}~~
	~~\end{pmatrix}~~.
	\\
	~~\end{array}~~
	~~</math>~~


	~~==See also==~~
	* [[Derrick Henry Lehmer]]
	* [[Hilbert matrix]]

	~~==References==~~
	* M. Newman and J. Todd, ''The evaluation of matrix inversion programs'', Journal of the Society for Industrial and Applied Mathematics, Volume 6, 1958, pages 466-476.

	~~[[Category:Matrices]]~~


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

en>Reatlas: Decapitalizing section headers.

2013-04-30T09:52:46Z

Decapitalizing section headers.

← Older revision		Revision as of 11:52, 30 April 2013
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	~~The writer~~'~~s name~~ is ~~Andera~~ and ~~she thinks it seems fairly great~~. ~~Ohio~~ is ~~exactly~~ where ~~my house~~ is ~~but my husband wants us to move. My working day occupation~~ is a ~~travel agent. To climb~~ is ~~something I truly appreciate performing.~~<br><br>~~My web site~~ ... ~~[http~~://~~www.publicpledge.com~~/~~blogs~~/~~post~~/~~7034 clairvoyants~~]		In [[mathematics]], particularly [[Matrix (mathematics)\|matrix theory]], the ''n×n'' '''Lehmer matrix''' (named after [[Derrick Henry Lehmer]]) is the constant [[symmetric matrix]] defined by
			:<math>A_{ij} =
			\begin{cases}
			i/j, & j\ge i \\
			j/i, & j<i.
			\end{cases}
			</math>

			Alternatively, this may be written as
			:<math>A_{ij} = \frac{\mbox{min}(i,j)}{\mbox{max}(i,j)}.</math>

			==Properties==

			As can be seen in the examples section, if ''A'' is an ''n×n'' Lehmer matrix and ''B'' is an ''m×m'' Lehmer matrix, then ''A'' is a [[submatrix]] of ''B'' whenever ''m''>''n''. The values of elements diminish toward zero away from the diagonal, where all elements have value 1.

			Interestingly, the [[matrix inverse\|inverse]] of a Lehmer matrix is a [[tridiagonal matrix]], where the [[superdiagonal]] and [[subdiagonal]] have strictly negative entries. Consider again the ''n×n'' ''A'' and ''m×m'' ''B'' Lehmer matrices, where ''m''>''n''. A rather peculiar property of their inverses is that ''A<sup>-1</sup>'' is ''nearly'' a submatrix of ''B<sup>-1</sup>'', except for the ''A<sub>n,n</sub>'' element, which is not equal to ''B<sub>m,m</sub>''.

			A Lehmer matrix of order ''n'' has [[trace of a matrix\|trace]] ''n''.

			==Examples==
			The 2×2, 3×3 and 4×4 Lehmer matrices and their inverses are shown below.
			:<math>
			\begin{array}{lllll}
			A_2=\begin{pmatrix}
			1 & 1/2 \\
			1/2 & 1
			\end{pmatrix};
			&
			A_2^{-1}=\begin{pmatrix}
			4/3 & -2/3 \\
			-2/3 & {\color{BrickRed}\mathbf{4/3}}
			\end{pmatrix};

			\\
			\\

			A_3=\begin{pmatrix}
			1 & 1/2 & 1/3 \\
			1/2 & 1 & 2/3 \\
			1/3 & 2/3 & 1
			\end{pmatrix};
			&
			A_3^{-1}=\begin{pmatrix}
			4/3 & -2/3 & \\
			-2/3 & 32/15 & -6/5 \\
			& -6/5 & {\color{BrickRed}\mathbf{9/5}}
			\end{pmatrix};

			\\
			\\

			A_4=\begin{pmatrix}
			1 & 1/2 & 1/3 & 1/4 \\
			1/2 & 1 & 2/3 & 1/2 \\
			1/3 & 2/3 & 1 & 3/4 \\
			1/4 & 1/2 & 3/4 & 1
			\end{pmatrix};
			&
			A_4^{-1}=\begin{pmatrix}
			4/3 & -2/3 & & \\
			-2/3 & 32/15 & -6/5 & \\
			& -6/5 & 108/35 & -12/7 \\
			& & -12/7 & {\color{BrickRed}\mathbf{16/7}}
			\end{pmatrix}.
			\\
			\end{array}
			</math>


			==See also==
			* [[Derrick Henry Lehmer]]
			* [[Hilbert matrix]]

			==References==
			* M. Newman and J. Todd, ''The evaluation of matrix inversion programs'', Journal of the Society for Industrial and Applied Mathematics, Volume 6, 1958, pages 466-476.

			[[Category:Matrices]]


			{{Linear-algebra-stub}}

en>NOrbeck: Removed redundant link

2012-07-18T18:44:04Z

Removed redundant link

New page

The writer's name is Andera and she thinks it seems fairly great. Ohio is exactly where my house is but my husband wants us to move. My working day occupation is a travel agent. To climb is something I truly appreciate performing.<br><br>My web site ... [http://www.publicpledge.com/blogs/post/7034 clairvoyants]