Vandermonde determinant

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)