en>Bender235: /* See also */

2014-06-10T21:10:08Z

See also

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	~~In [[linear algebra]], an '''alternant matrix''', is a [[matrix (math)\|matrix]] with a particular structure, in which successive columns have a particular function applied to their entries.~~ ~~An '''alternant determinant''' is the~~ [~~[determinant]] of an alternant matrix. Such a matrix of size ''m'' × ''n'' matrix may be written out as~~

	:~~<math>M=\begin{bmatrix}~~
	~~f_1(\alpha_1) & f_2(\alpha_1) & \dots & f_n(\alpha_1)\\~~
	~~f_1(\alpha_2) & f_2(\alpha_2) & \dots & f_n(\alpha_2)\\~~
	~~f_1(\alpha_3) & f_2(\alpha_3) & \dots & f_n(\alpha_3)\\~~
	~~\vdots & \vdots & \ddots &\vdots \\~~
	~~f_1(\alpha_m) & f_2(\alpha_m) & \dots & f_n(\alpha_m)\\~~
	~~\end{bmatrix}<~~/~~math>~~
	~~or more succinctly~~
	~~:<math>M_{i,j} = f_j(\alpha_i)<~~/~~math>~~
	~~for all indices ''i'' and ''j''~~. ~~(Some authors use the [[transpose]] of the above matrix~~.)

	~~Examples of alternant matrices include [[Vandermonde matrix\|Vandermonde matrices]], for which <math>f_i(\alpha)=\alpha^{i-1}<~~/~~math> and [[Moore matrices]] for which <math>f_i(\alpha)=\alpha^{q^{i-1}}<~~/~~math>~~.

	~~If <math>n~~ = ~~m</math> and the <math>f_j(x)</math> functions are all polynomials we have some additional results: if <math>\alpha_i~~ = ~~\alpha_j</math> for any <math>i < j</math> then the determinant of any alternant matrix is zero (as a row~~ is ~~then repeated), thus <math>(\alpha_j - \alpha_i)</math> divides~~ the ~~determinant for all <math>1 \leq i < j \leq n</math>.~~ ~~As such, if we take~~
	:~~<math>~~
	~~V = \begin{bmatrix}~~
	~~1 & \alpha_1 & \dots & \alpha_1^{n-1} \\~~
	~~1 & \alpha_2 & \dots & \alpha_2^{n-1} \\~~
	~~1 & \alpha_3 & \dots & \alpha_3^{n-1} \\~~
	~~\vdots & \vdots & \ddots &\vdots \\~~
	~~1 & \alpha_n & \dots & \alpha_n^{n-1} \\~~
	~~\end{bmatrix}~~
	</~~math>~~
	~~(a [[Vandermonde matrix]]) then <math>\prod_{i < j} (\alpha_j - \alpha_i) = \det V<~~/~~math> divides such polynomial alternant determinants~~. ~~The ratio <math>\frac{\det M}{\det V}<~~/~~math>~~ is ~~called a bialternant~~. ~~In the case~~ where ~~each function~~ <~~math~~>~~f_j(x) = x^{m_j}~~<~~/math~~>~~, this forms the classical definition of the [[Schur polynomial]]s.~~

	~~Alternant matrices are used in [[coding theory]] in the construction of [[alternant code]]s.~~

	~~==See also==~~
	* [[List of matrices]]

	~~==References==~~
	* {{cite book \| author=Thomas Muir \| authorlink=Thomas Muir (~~mathematician) \| title=A treatise on the theory of determinants \| date=1960 \| publisher=~~[~~[Dover Publications]] \| pages=321–363 }}~~
	* {{cite book \| author=A. C. ~~Aitken \| authorlink=Alexander Aitken \| title=Determinants and Matrices \| date=1956 \| publisher=Oliver and Boyd Ltd \| pages=111–123 }}~~
	* {{cite book \| author=Richard P. ~~Stanley \| authorlink=Richard P~~. ~~Stanley \| title=Enumerative Combinatorics \| date=1999 \| publisher=[[Cambridge University Press~~]~~] \| pages=334–342 }}~~

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

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

141.35.244.170: /* See also */

2013-11-06T11:35:58Z

en>Melcombe: /* External links */ remove section as all links dead, add {{noreferences}}

2012-02-16T22:32:20Z

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en>Melcombe: /* External links */ remove section as all links dead, add {{noreferences}}