en>Marcello donatelli: /* Other models */

2014-01-05T20:05:51Z

Other models

← Older revision		Revision as of 22:05, 5 January 2014
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	~~Emilia Shryock~~ is ~~my name but you can contact me something you like~~. ~~Body developing~~ is ~~one~~ of the ~~things I adore most~~. ~~North Dakota~~ is ~~her beginning location but she~~ will have to ~~transfer one day~~ or ~~an additional. Hiring is my profession.~~<br><br>~~Also visit my website ...~~ [~~http://i4p~~.~~info/article~~.~~php?~~id=~~114136 std testing at home~~]		In [[mathematics]], an '''anti-diagonal matrix''' is a [[matrix (mathematics)\|matrix]] where all the entries are zero except those on the diagonal going from the lower left corner to the upper right corner (↗), known as the anti-diagonal.

			More precisely, an ''n''-by-''n'' matrix ''A'' is an anti-diagonal matrix if the (''i'', ''j'') element is zero for all ''i'', ''j'' ∈ {1, …, ''n''} with ''i'' + ''j'' ≠ ''n'' + 1.

			An example of an anti-diagonal matrix is
			:<math> \begin{bmatrix} 0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 5 & 0 & 0 \\ 0 & 7 & 0 & 0 & 0 \\ -1 & 0 & 0 & 0 & 0 \end{bmatrix}. </math>

			All anti-diagonal matrices are also [[persymmetric matrix\|persymmetric]].

			The product of two anti-diagonal matrices is a [[diagonal matrix]]. Furthermore, the product of an anti-diagonal matrix with a diagonal matrix is anti-diagonal, as is the product of a diagonal matrix with an anti-diagonal matrix.

			An anti-diagonal matrix is [[invertible]] if and only if the entries on the diagonal from the lower left corner to the upper right corner are nonzero. The inverse of any invertible anti-diagonal matrix is also anti-diagonal, as can be seen from the paragraph above. The [[determinant]] of an anti-diagonal matrix has [[absolute value]] given by the [[product (mathematics)\|product]] of the entries on the diagonal from the lower left corner to the upper right corner. However, the sign of this determinant will vary because the one nonzero signed elementary product from an anti-diagonal matrix will have a different sign depending on whether the [[permutation]] related to it is odd or even:

			{\| class="wikitable"
			!Matrix size
			!Permutation for<br>nonzero elementary product of<br>anti-diagonal matrix
			!Even or odd
			!Sign of elementary product
			\|-
			\|2 × 2
			\|{2, 1}
			\|Odd
			\| '''-'''
			\|-
			\|3 × 3
			\|{3, 2, 1}
			\|Odd
			\| '''-'''
			\|-
			\|4 × 4
			\|{4, 3, 2, 1}
			\|Even
			\| '''+'''
			\|-
			\|5 × 5
			\|{5, 4, 3, 2, 1}
			\|Even
			\| '''+'''
			\|-
			\|6 × 6
			\|{6, 5, 4, 3, 2, 1}
			\|Odd
			\| '''-'''
			\|-
			\|}

			More precisely, the sign of the elementary product needed to calculate the determinant of an anti-diagonal matrix is related to whether the corresponding [[triangular number]] is even or odd. This is because the number of inversions in the permutation for the only nonzero signed elementary product of any ''n'' × ''n'' anti-diagonal matrix is always equal to the ''n''th such number.

			==External links==
			* {{planetmath reference\|id=6964\|title=Anti-diagonal matrix}}



			[[Category:Sparse matrices]]

en>Ketiltrout: dab

2012-08-14T09:20:31Z

dab

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Emilia Shryock is my name but you can contact me something you like. Body developing is one of the things I adore most. North Dakota is her beginning location but she will have to transfer one day or an additional. Hiring is my profession.<br><br>Also visit my website ... [http://i4p.info/article.php?id=114136 std testing at home]

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en>Marcello donatelli: /* Other models */

en>Ketiltrout: dab