范德蒙德矩阵行列式 & 循环矩阵行列式的证明

范德蒙德矩阵的行列式

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

Proof:

\[\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}
\]

循环矩阵的行列式

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

Proof:

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