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| {{distinguish|alternating sign matrix}}
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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
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| :<math>M=\begin{bmatrix} | |
| f_1(\alpha_1) & f_2(\alpha_1) & \dots & f_n(\alpha_1)\\
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| f_1(\alpha_2) & f_2(\alpha_2) & \dots & f_n(\alpha_2)\\
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| f_1(\alpha_3) & f_2(\alpha_3) & \dots & f_n(\alpha_3)\\
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| \vdots & \vdots & \ddots &\vdots \\
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| f_1(\alpha_m) & f_2(\alpha_m) & \dots & f_n(\alpha_m)\\
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| \end{bmatrix}</math>
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| or more succinctly
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| :<math>M_{i,j} = f_j(\alpha_i)</math>
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| for all indices ''i'' and ''j''. (Some authors use the [[transpose]] of the above matrix.)
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| 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>.
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| 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
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| :<math> | |
| V = \begin{bmatrix}
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| 1 & \alpha_1 & \dots & \alpha_1^{n-1} \\
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| 1 & \alpha_2 & \dots & \alpha_2^{n-1} \\
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| 1 & \alpha_3 & \dots & \alpha_3^{n-1} \\
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| \vdots & \vdots & \ddots &\vdots \\
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| 1 & \alpha_n & \dots & \alpha_n^{n-1} \\
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| \end{bmatrix}
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| </math>
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| (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.
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| Alternant matrices are used in [[coding theory]] in the construction of [[alternant code]]s.
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| ==See also==
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| * [[List of matrices]]
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| ==References==
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| * {{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 }}
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| * {{cite book | author=A. C. Aitken | authorlink=Alexander Aitken | title=Determinants and Matrices | date=1956 | publisher=Oliver and Boyd Ltd | pages=111–123 }}
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| * {{cite book | author=Richard P. Stanley | authorlink=Richard P. Stanley | title=Enumerative Combinatorics | date=1999 | publisher=[[Cambridge University Press]] | pages=334–342 }}
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| [[Category:Matrices]]
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| [[Category:Determinants]]
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| {{Linear-algebra-stub}}
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