純粋・応用数学・数学隣接分野（含むガロア理論）12

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544: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2023/01/08(日) 17:03:17.52 ID:9zXu/9tz

>>530
>　例えば、線型代数でファンデルモンド行列って必ず出てくるけど
>　あれがいったいどうして出てきたのか、そういうことはまず語られない

あんたの昭和数学科時代に、
いまみたく検索が充実していれば、大分ちがったろうね

・ファンデルモンド行列式は、根の差積 det(V)=Π_1<= i<j<= n (xj-xi)を与えるのか（説明は日本語版にある）
・Vandermonde氏は、Vandermonde was a violinist, and became engaged with mathematics only around 1770 とある。多分35歳から数学でそれ以前はバイオリン？ｗ
・”The Vandermonde determinant does not make an explicit appearance.”とあるけどｗ
・因みに、周知だが、行列式 determinantの方が数学史的には早くて、行列 matrix の概念は後から
・フランス語も見たけど、仏語の”Demonstration”が良いかも（英語版にはない）
・The discrete Fourier transform is defined by a specific Vandermonde matrix とあるね

https://ja.wikipedia.org/wiki/%E3%83%B4%E3%82%A1%E3%83%B3%E3%83%87%E3%83%AB%E3%83%A2%E3%83%B3%E3%83%89%E3%81%AE%E8%A1%8C%E5%88%97%E5%BC%8F
ヴァンデルモンドの行列式
https://fr.wikipedia.org/wiki/Matrice_de_Vandermonde
Matrice de Vandermonde
https://en.wikipedia.org/wiki/Vandermonde_matrix
Vandermonde matrix
The determinant of a square Vandermonde matrix is called a Vandermonde polynomial or Vandermonde determinant. Its value is the polynomial
det(V)=Π_1<= i<j<= n (xj-xi)
which is non-zero if and only if all xi are distinct.

It thus depends on the choice of an order for the xi, while its square, the discriminant, does not depend on any order, and this implies, by Galois theory, that the discriminant is a polynomial function of the coefficients of the polynomial that has the {\displaystyle xi as roots.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1671460269/544

545: 現代数学の系譜 雑談 ◆yH25M02vWFhP  [] 2023/01/08(日) 17:04:02.18 ID:9zXu/9tz

つづき

Applications
The Vandermonde determinant is used in the representation theory of the symmetric group.[8]

When the values αk belong to a finite field, then the Vandermonde determinant is also called a Moore determinant and has specific properties that are used, for example, in the theory of BCH code and Reed?Solomon error correction codes.

The discrete Fourier transform is defined by a specific Vandermonde matrix, the DFT matrix, where the numbers αi are chosen to be roots of unity. Using the Fast Fourier Transform it is possible to compute the product of a Vandermonde matrix with a vector in O(n(log n)^2) time.[9]

https://en.wikipedia.org/wiki/Alexandre-Th%C3%A9ophile_Vandermonde
Alexandre-Theophile Vandermonde (28 February 1735 ? 1 January 1796)
Biography
Vandermonde was a violinist, and became engaged with mathematics only around 1770. In Memoire sur la resolution des equations (1771) he reported on symmetric functions and solution of cyclotomic polynomials; this paper anticipated later Galois theory (see also abstract algebra for the role of Vandermonde in the genesis of group theory).
The same year he was elected to the French Academy of Sciences. Memoire sur des irrationnelles de differents ordres avec une application au cercle (1772) was on combinatorics, and Memoire sur l'elimination (1772) on the foundations of determinant theory.
The Vandermonde determinant does not make an explicit appearance.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1671460269/545

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