范德蒙德矩阵行列式 & 循环矩阵行列式的证明

\[\begin{vmatrix} 1 & 1 & 1 & \dots & 1 \\ x_1 & x_2 & x_3 & \dots & x_n \\ x_1^2 & x_2^2 & x_3^2 & \dots & x_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \dots & x_n^{n-1} \\ \end{vmatrix} =\prod\limits_{i>j}(x_i-x_j) \]

\[\begin{aligned} & \begin{vmatrix} 1 & 1 & 1 & \dots & 1 \\ x_1 & x_2 & x_3 & \dots & x_n \\ x_1^2 & x_2^2 & x_3^2 & \dots & x_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1^{n-2} & x_2^{n-2} & x_3^{n-2} & \dots & x_n^{n-2} \\ x_1^{n-1} & x_2^{n-1} & x_3^{n-1} & \dots & x_n^{n-1} \\ \end{vmatrix} \\ \\ =& \begin{vmatrix} 1 & 1 & 1 & \dots & 1 \\ x_1 & x_2 & x_3 & \dots & x_n \\ x_1^2 & x_2^2 & x_3^2 & \dots & x_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1^{n-2} & x_2^{n-2} & x_3^{n-2} & \dots & x_n^{n-2} \\ x_1^{n-1}-x_1x_1^{n-2} & x_2^{n-1}-x_1x_2^{n-2} & x_3^{n-1}-x_1x_3^{n-2} & \dots & x_n^{n-1}-x_1x_n^{n-2} \\ \end{vmatrix} \texttt{（用第 n-1 行乘 x1 去减第 n 行）} \\ \\ =& \begin{vmatrix} 1 & 1 & 1 & \dots & 1 \\ x_1 & x_2 & x_3 & \dots & x_n \\ x_1^2 & x_2^2 & x_3^2 & \dots & x_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1^{n-2}-x_1x_1^{n-3} & x_2^{n-2}-x_1x_2^{n-3} & x_3^{n-2}-x_1x_3^{n-3} & \dots & x_n^{n-2}-x_1x_n^{n-3} \\ x_1^{n-1}-x_1x_1^{n-2} & x_2^{n-1}-x_1x_2^{n-2} & x_3^{n-1}-x_1x_3^{n-2} & \dots & x_n^{n-1}-x_1x_n^{n-2} \\ \end{vmatrix} \texttt{（用第 n-2 行乘 x1 去减第 n-1 行）} \\ \\ =&\dots\\ =& \begin{vmatrix} 1 & 1 & 1 & \dots & 1 \\ x_1-x_1 & x_2-x_1 & x_3-x_1 & \dots & x_n-x_1 \\ x_1^2-x_1x_1 & x_2^2-x_1x_2 & x_3^2-x_1x_3 & \dots & x_n^2-x_1x_n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ x_1^{n-2}-x_1x_1^{n-3} & x_2^{n-2}-x_1x_2^{n-3} & x_3^{n-2}-x_1x_3^{n-3} & \dots & x_n^{n-2}-x_1x_n^{n-3} \\ x_1^{n-1}-x_1x_1^{n-2} & x_2^{n-1}-x_1x_2^{n-2} & x_3^{n-1}-x_1x_3^{n-2} & \dots & x_n^{n-1}-x_1x_n^{n-2} \\ \end{vmatrix} \texttt{（以此类推）} \\ \\ =& \begin{vmatrix} 1 & 1 & 1 & \dots & 1 \\ 0 & x_2-x_1 & x_3-x_1 & \dots & x_n-x_1 \\ 0 & x_2^2-x_1x_2 & x_3^2-x_1x_3 & \dots & x_n^2-x_1x_n \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & x_2^{n-2}-x_1x_2^{n-3} & x_3^{n-2}-x_1x_3^{n-3} & \dots & x_n^{n-2}-x_1x_n^{n-3} \\ 0 & x_2^{n-1}-x_1x_2^{n-2} & x_3^{n-1}-x_1x_3^{n-2} & \dots & x_n^{n-1}-x_1x_n^{n-2} \\ \end{vmatrix} \\ \\ =& \begin{vmatrix} x_2-x_1 & x_3-x_1 & \dots & x_n-x_1 \\ x_2^2-x_1x_2 & x_3^2-x_1x_3 & \dots & x_n^2-x_1x_n \\ \vdots & \vdots & \ddots & \vdots \\ x_2^{n-2}-x_1x_2^{n-3} & x_3^{n-2}-x_1x_3^{n-3} & \dots & x_n^{n-2}-x_1x_n^{n-3} \\ x_2^{n-1}-x_1x_2^{n-2} & x_3^{n-1}-x_1x_3^{n-2} & \dots & x_n^{n-1}-x_1x_n^{n-2} \\ \end{vmatrix} \\ \\ =& (x_2-x_1)(x_3-x_1)\dots(x_n-x_1) \begin{vmatrix} 1 & 1 & \dots & 1 \\ x_2 & x_3 & \dots & x_n \\ x_2^2 & x_3^2 & \dots & x_n^2 \\ \vdots & \vdots & \ddots & \vdots \\ x_2^{n-2} & x_3^{n-2} & \dots & x_n^{n-2} \\ x_2^{n-1} & x_3^{n-1} & \dots & x_n^{n-1} \\ \end{vmatrix} \texttt{（提出每列的公因式）} \\ \\ =&\dots\\ \\ =&\prod\limits_{i>j}(x_i-x_j) \end{aligned} \]

\[A= \begin{pmatrix} a_1 & a_2 & a_3 & \dots & a_n \\ a_n & a_1 & a_2 & \dots & a_{n-1} \\ a_{n-1} & a_n & a_1 & \dots & a_{n-2} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ a_2 & a_3 & a_4 & \dots & a_1 \\ \end{pmatrix} \\ \texttt{ Let }f(x)=a_1+a_2x+a_3x^2+\dots+a_nx^{n-1} \\ \texttt{Then } |A|=f(\epsilon_1)f(\epsilon_2)\dots f(\epsilon_n) \\ \texttt{其中 }\epsilon_i \texttt{ 是 1 的 n 个互不相同的 n 次单位根} \]

\[\texttt{Let } V= \begin{pmatrix} 1 & 1 & 1 & \dots & 1 \\ \epsilon_1 & \epsilon_2 & \epsilon_3 & \dots & \epsilon_n \\ \epsilon_1^2 & \epsilon_2^2 & \epsilon_3^2 & \dots & \epsilon_n^2 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \epsilon_1^{n-1} & \epsilon_2^{n-1} & \epsilon_3^{n-1} & \dots & \epsilon_n^{n-1} \\ \end{pmatrix} \\ \texttt{Then } AV= \begin{pmatrix} f(\epsilon_1) & f(\epsilon_2) & f(\epsilon_3) & \dots & f(\epsilon_n) \\ \epsilon_1f(\epsilon_1) & \epsilon_2f(\epsilon_2) & \epsilon_3f(\epsilon_3) & \dots & \epsilon_nf(\epsilon_n) \\ \epsilon_1^2f(\epsilon_1) & \epsilon_2^2f(\epsilon_2) & \epsilon_3^2f(\epsilon_3) & \dots & \epsilon_n^2f(\epsilon_n) \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ \epsilon_1^{n-1}f(\epsilon_1) & \epsilon_2^{n-1}f(\epsilon_2) & \epsilon_3^{n-1}f(\epsilon_3) & \dots & \epsilon_n^{n-1}f(\epsilon_n) \\ \end{pmatrix} \\ \therefore |AV|=f(\epsilon_1)f(\epsilon_2)\dots f(\epsilon_n)|V|\\ |A|=f(\epsilon_1)f(\epsilon_2)\dots f(\epsilon_n) \]