范德蒙德行列式的推導(dǎo)

2023-06-03 22:01 作者:~Sakuno醬 0人讀過(guò) | 我要投稿

命題

證明? $%5Cbegin%7Bvmatrix%7D%0A1%20%26%20x_1%20%26%20x_1%5E2%20%26%20..%20%26x_1%5E%7Bn-1%7D%20%5C%5C%0A1%20%26%20x_2%20%26%20x_2%5E2%20%26%20..%20%26%20x_2%5E%7Bn-1%7D%20%5C%5C%0A..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%5C%5C%0A1%20%26%20x_n%20%26%20x_n%5E2%20%26%20..%20%26%20x_n%5E%7Bn-1%7D%20%0A%5Cend%7Bvmatrix%7D$ 等于? $%5Cprod_%7B1%20%5Cle%20i%20%3C%20j%20%5Cle%20n%7D%20(x_j-x_i)$

范德蒙德行列式的特點(diǎn)就是第 $k$ 列的次數(shù)是 $k-1$ 次，每一列是齊次的

我們的思路是作初等行變換，同時(shí)希望改變后的每一列都是齊次的。

對(duì) $n$ 使用歸納法

假設(shè)? $n%3Dk$ 時(shí)成立

考慮 $n%3Dk%2B1$

$%5Cbegin%7Bvmatrix%7D%0A1%20%26%20x_1%20%26%20x_1%5E2%20%26%20..%20%26%20x_1%5E%7Bk-1%7D%20%260%20%5C%5C%0A1%20%26%20x_2%20%26%20x_2%5E2%20%26%20..%20%26%20x_2%5E%7Bk-1%7D%20%26%20x_2%5E%7Bk%7D%20-x_1x_2%5E%7Bk-1%7D%20%5C%5C%0A..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%5C%5C%0A1%20%26%20x_%7Bk%2B1%7D%20%26%20x_%7Bk%2B1%7D%5E2%20%26%20..%20%26%20x_k%5E%7Bk-1%7D%20%26%20x_%7Bk%2B1%7D%5E%7Bk%7D%20-%20x_1x_%7Bk%7D%5E%7Bk-1%7D%0A%5Cend%7Bvmatrix%7D$

作初等列變換, 把第 $k$ 列乘以? $-x_1$ 加到 $k%2B1$ 列

然后是第 $k-1$ 列乘以 $-x_1$ 加到 $k$ 列，繼續(xù)重復(fù) $k$ 次，得到

$%5Cbegin%7Bvmatrix%7D%0A1%20%26%200%20%26%200%20%26%20..%20%26%200%20%260%20%5C%5C%0A1%20%26%20x_2-x_1%20%26%20x_2%5E2%20-x_1x_2%26%20..%20%26%20x_2%5E%7Bk-1%7D-x_1x_2%5E%7Bk-2%7D%20%26%20x_2%5E%7Bk%7D%20-x_1x_2%5E%7Bk-1%7D%20%5C%5C%0A..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%5C%5C%0A1%20%26%20x_%7Bk%2B1%7D%20-x_1%26%20x_%7Bk%2B1%7D%5E2%20-x_%7B1%7Dx_%7Bk%2B1%7D%26%20..%20%26%20x_k%5E%7Bk-1%7D%20-x_1x_k%5E%7Bk-2%7D%26%20x_%7Bk%2B1%7D%5E%7Bk%7D%20-%20x_1x_%7Bk%7D%5E%7Bk-1%7D%0A%5Cend%7Bvmatrix%7D$

根據(jù)行列式的性質(zhì)按第一行展開(kāi)得到

$%5Cbegin%7Bvmatrix%7D%0A%20x_2-x_1%20%26%20x_2%5E2%20-x_1x_2%26%20..%20%26%20x_2%5E%7Bk-1%7D-x_1x_2%5E%7Bk-2%7D%20%26%20x_2%5E%7Bk%7D%20-x_1x_2%5E%7Bk-1%7D%20%5C%5C%0A%20x_3-x_1%20%26%20x_3%5E2%20-x_1x_3%26%20..%20%26%20x_3%5E%7Bk-1%7D-x_1x_3%5E%7Bk-2%7D%20%26%20x_3%5E%7Bk%7D%20-x_1x_3%5E%7Bk-1%7D%20%5C%5C%0A..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%5C%5C%0A%20x_%7Bk%2B1%7D%20-x_1%26%20x_%7Bk%2B1%7D%5E2%20-x_%7B1%7Dx_%7Bk%2B1%7D%26%20..%20%26%20x_k%5E%7Bk-1%7D%20-x_1x_k%5E%7Bk-2%7D%26%20x_%7Bk%2B1%7D%5E%7Bk%7D%20-%20x_1x_%7Bk%7D%5E%7Bk-1%7D%0A%5Cend%7Bvmatrix%7D$

每一行可以提取公因子? $x_2-x_1$ ， $x_3-x_1$ ?等

$(x_2-x_1)(x_3-x_1)..(x_%7Bk%2B1%7D-x_1)%5Cbegin%7Bvmatrix%7D%0A%201%20%26%20x_2%26%20..%20%26%20x_2%5E%7Bk-2%7D%20%26%20x_2%5E%7Bk-1%7D%20%5C%5C%0A%201%20%26%20x_3%20%26%20..%20%26%20x_3%5E%7Bk-2%7D%20%26%20x_3%5E%7Bk-1%7D%20%5C%5C%0A..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%5C%5C%0A%201%26%20x_%7Bk%2B1%7D%26%20..%20%26%20x_k%5E%7Bk-2%7D%26%20x_%7Bk%7D%5E%7Bk-1%7D%0A%5Cend%7Bvmatrix%7D$