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A [[Determinant|determinant]] of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960601.png" /> of the type
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$#C+1 = 11 : ~/encyclopedia/old_files/data/V096/V.0906060 Vandermonde determinant
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960602.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960603.png" /> are elements of a commutative ring. For any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960604.png" />,
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A [[Determinant|determinant]] of order  $  n $
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of the type
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960605.png" /></td> </tr></table>
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$$ \tag{* }
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B ( a _ {1} \dots a _ {n} )  = \
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\left |
  
If the ring has no zero divisors, the fundamental property of a Vandermonde determinant holds: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960606.png" /> if and only if not all the elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960607.png" /> are different from each other. The determinant was first studied by A.T. Vandermonde for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960608.png" /> , and then in 1815 by A.L. Cauchy .
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where  $  a _ {1} \dots a _ {n} $
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are elements of a commutative ring. For any  $  n \geq  2 $,
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$$
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B ( a _ {1} \dots a _ {n} )  = \
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\prod _ {1 \leq  j < i \leq  n } ( a _ {i} - a _ {j} ).
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$$
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If the ring has no zero divisors, the fundamental property of a Vandermonde determinant holds: $  B( a _ {1} \dots a _ {n} ) = 0 $
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if and only if not all the elements $  a _ {1} \dots a _ {n} $
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are different from each other. The determinant was first studied by A.T. Vandermonde for the case $  n = 3 $,  
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and then in 1815 by A.L. Cauchy .
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  A.T. Vandermonde,  ''Histoire Acad. R. Sci. Paris''  (1771 (1774))  pp. 365–416</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  A.T. Vandermonde,  ''Histoire Acad. R. Sci. Paris''  (1772 (1776))  pp. 516–532</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  A.A. Cauchy,  "Mémoire sur les fonctions qui ne peuvent obtenir que deux values"  ''J. École Polytechnique'' , '''17''' :  10  (1815)  pp. 29-</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  A.L. Cauchy,  "Mémoire sur les fonctions qui ne peuvent obtenir que deux values" , ''Oeuvres Sér. 2'' , '''1''' , Gauthier-Villars  (1905)  pp. 91–169</TD></TR></table>
 
<table><TR><TD valign="top">[1a]</TD> <TD valign="top">  A.T. Vandermonde,  ''Histoire Acad. R. Sci. Paris''  (1771 (1774))  pp. 365–416</TD></TR><TR><TD valign="top">[1b]</TD> <TD valign="top">  A.T. Vandermonde,  ''Histoire Acad. R. Sci. Paris''  (1772 (1776))  pp. 516–532</TD></TR><TR><TD valign="top">[2a]</TD> <TD valign="top">  A.A. Cauchy,  "Mémoire sur les fonctions qui ne peuvent obtenir que deux values"  ''J. École Polytechnique'' , '''17''' :  10  (1815)  pp. 29-</TD></TR><TR><TD valign="top">[2b]</TD> <TD valign="top">  A.L. Cauchy,  "Mémoire sur les fonctions qui ne peuvent obtenir que deux values" , ''Oeuvres Sér. 2'' , '''1''' , Gauthier-Villars  (1905)  pp. 91–169</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
The matrix
 
The matrix
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v0960609.png" /></td> </tr></table>
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$$
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\left (
  
 
participating in (*) is called a Vandermonde matrix.
 
participating in (*) is called a Vandermonde matrix.
  
The Vandermonde matrix plays a role in approximation theory. E.g., using it one can prove that there is a unique polynomial of degree <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v09606010.png" /> taking prescribed values at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/v/v096/v096060/v09606011.png" /> distinct points, cf. [[#References|[a1]]], p. 58. See [[#References|[a1]]], p. 64, Problem 13, for an algorithm to compute the inverse of a Vandermonde matrix.
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The Vandermonde matrix plays a role in approximation theory. E.g., using it one can prove that there is a unique polynomial of degree $  n $
 +
taking prescribed values at $  n+ 1 $
 +
distinct points, cf. [[#References|[a1]]], p. 58. See [[#References|[a1]]], p. 64, Problem 13, for an algorithm to compute the inverse of a Vandermonde matrix.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.W. Cheney,  "Introduction to approximation theory" , Chelsea, reprint  (1982)</TD></TR></table>

Latest revision as of 08:27, 6 June 2020


A determinant of order $ n $ of the type

$$ \tag{* } B ( a _ {1} \dots a _ {n} ) = \ \left | where $ a _ {1} \dots a _ {n} $ are elements of a commutative ring. For any $ n \geq 2 $, $$ B ( a _ {1} \dots a _ {n} ) = \ \prod _ {1 \leq j < i \leq n } ( a _ {i} - a _ {j} ). $$ If the ring has no zero divisors, the fundamental property of a Vandermonde determinant holds: $ B( a _ {1} \dots a _ {n} ) = 0 $ if and only if not all the elements $ a _ {1} \dots a _ {n} $ are different from each other. The determinant was first studied by A.T. Vandermonde for the case $ n = 3 $, and then in 1815 by A.L. Cauchy . ===='"`UNIQ--h-0--QINU`"'References==== <table><tr><td valign="top">[1a]</td> <td valign="top"> A.T. Vandermonde, ''Histoire Acad. R. Sci. Paris'' (1771 (1774)) pp. 365–416</td></tr><tr><td valign="top">[1b]</td> <td valign="top"> A.T. Vandermonde, ''Histoire Acad. R. Sci. Paris'' (1772 (1776)) pp. 516–532</td></tr><tr><td valign="top">[2a]</td> <td valign="top"> A.A. Cauchy, "Mémoire sur les fonctions qui ne peuvent obtenir que deux values" ''J. École Polytechnique'' , '''17''' : 10 (1815) pp. 29-</td></tr><tr><td valign="top">[2b]</td> <td valign="top"> A.L. Cauchy, "Mémoire sur les fonctions qui ne peuvent obtenir que deux values" , ''Oeuvres Sér. 2'' , '''1''' , Gauthier-Villars (1905) pp. 91–169</td></tr></table> ===='"`UNIQ--h-1--QINU`"'Comments==== The matrix $$ \left (

participating in (*) is called a Vandermonde matrix.

The Vandermonde matrix plays a role in approximation theory. E.g., using it one can prove that there is a unique polynomial of degree $ n $ taking prescribed values at $ n+ 1 $ distinct points, cf. [a1], p. 58. See [a1], p. 64, Problem 13, for an algorithm to compute the inverse of a Vandermonde matrix.

References

[a1] E.W. Cheney, "Introduction to approximation theory" , Chelsea, reprint (1982)
How to Cite This Entry:
Vandermonde determinant. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Vandermonde_determinant&oldid=49108
This article was adapted from an original article by V.N. Remeslennikov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article