范德蒙德行列式的推导

2023-06-03 22:01 作者:~Sakuno酱 0人读过 | 我要投稿

命题

证明 $%5Cbegin%7Bvmatrix%7D%0A1%20%26%20x_1%20%26%20x_1%5E2%20%26%20..%20%26x_1%5E%7Bn-1%7D%20%5C%5C%0A1%20%26%20x_2%20%26%20x_2%5E2%20%26%20..%20%26%20x_2%5E%7Bn-1%7D%20%5C%5C%0A..%20%26%20..%20%26%20..%20%26%20..%20%26%20..%20%5C%5C%0A1%20%26%20x_n%20%26%20x_n%5E2%20%26%20..%20%26%20x_n%5E%7Bn-1%7D%20%0A%5Cend%7Bvmatrix%7D$ 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点就是第 $k$ 列的次数是 $k-1$ 次，每一列是齐次的

我们的思路是作初等行变换，同时希望改变后的每一列都是齐次的。

对 $n$ 使用归纳法

假设 $n%3Dk$ 时成立

考虑 $n%3Dk%2B1$

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

作初等列变换, 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，继续重复 $k$ 次，得到

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

根据行列式的性质按第一行展开得到

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

每一行可以提取公因子 $x_2-x_1$ ， $x_3-x_1$ 等

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

注意到右边的行列式是 $k$ 阶的

$%3D(x_2-x_1)(x_3-x_1)..(x_%7Bk%2B1%7D-x_1)%5Cprod_%7B2%5Cle%20i%3Cj%5Cle%20k%2B1%7D(x_j-x_i)$

$%3D%5Cprod_%7B1%20%5Cle%20i%20%3Cj%20%5Cle%20k%2B1%7D(x_j-x_i)$

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范德蒙德行列式的推导

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