一个类范德蒙矩阵的行列式值的符号

      由博文“一个类范德蒙矩阵的行列式值估计”(http://blog.sciencenet.cn/home.php?mod=space&uid=3343777&do=blog&id=1078764)有如下类范德蒙矩阵的行列式值的符号估计。

      【引理】 (1) 定义类范德蒙矩阵的行列式值如下

$F_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{0,k_{2},k_{3},\cdots,k_{n}}=\det\left(\left[\begin{array}{cccc} 1 & \lambda_{1}^{k_{2}} & \cdots & \lambda_{1}^{k_{n}}\\ 1 & \lambda_{2}^{k_{2}} & \cdots & \lambda_{2}^{k_{n}}\\ \vdots & \vdots & \ddots & \vdots\\ 1 & \lambda_{n}^{k_{2}} & \cdots & \lambda_{n}^{k_{n}} \end{array}\right]\right)$

则对满足 $0\leq k_{2}0(\lambda_{i}<0),i=\overline{1,n}$ 的任意 $k_{i$ }和 $\lambda_{i}\left(i=\overline{1,n}\right)$ , $F_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{0,k_{2},k_{3},\cdots,k_{n}}$ 取相同的符号；

      (2) 若进一步有0 $<\lambda_{1}<\lambda_{2}<\cdots<\lambda_{n}$ ，则 $F_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{0,k_{2},k_{3},\cdots,k_{n}}$ 取正号，即

$F_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{0,k_{2},k_{3},\cdots,k_{n}}>0$

若 $\lambda_{1}<\lambda_{2}<\cdots<\lambda_{n}<0$ ，则 $F_{\lambda_{1},\lambda_{2},\cdots,\lambda_{n}}^{0,k_{2},k_{3},\cdots,k_{n}}>0" style="text-align:center;$ 的符号为 $(-1)^{\sum_{j=2}^{n}\left(k_{j}-j+1\right)}$ .