一般的n阶范德蒙行列式计算的两个主要步骤
一般的n阶范德蒙行列式计算的两个主要步骤
一列(行)化为零→按列(行)提取公因式一列(行)化为零 \rightarrow 按列(行)提取公因式一列(行)化为零→按列(行)提取公因式
Dn=∣111x1x2x3x12x22x32∣D_n=\left|\begin{array}{cccc} 1& 1 &1 \\ x_1&{ x_2}&{ x_3}\\ { x_1}^{2}&{ x_2}^{2}&{ x_3}^{2}\\ \end{array}\right| Dn=∣∣∣∣∣∣1x1x121x2x221x3x32∣∣∣∣∣∣
一列(行)化为零(贪心算法)一列(行)化为零(贪心算法)一列(行)化为零(贪心算法)
Dn=∣1110x2−x1x3−x10x22−x2x1x32−x3x1∣=1∗∣B∣对于B=∣x2−x1x3−x1x22−x2x1x32−x3x1∣按列提取公因式则B=(x2−x1)(x3−x1)∣11x2x3∣D_n=\left|\begin{array}{cccc} 1& 1 &1 \\ 0&{ x_2-x_1}&{ x_3-x_1}\\ { 0}&{ x_2}^{2}-x_2x_1&{ x_3}^{2}-x_3x_1\\ \end{array}\right| \\ =1*|B|\\ 对于B=\left|\begin{array}{cccc} { x_2-x_1}&{ x_3-x_1}\\ { x_2}^{2}-x_2x_1&{ x_3}^{2}-x_3x_1\\ \end{array}\right| \\ 按列提取公因式\\ 则B=( x_2-x_1)(x_3-x_1)\left|\begin{array}{cccc} { 1}&{1}\\ { x_2}&{ x_3}\end{array}\right| Dn=∣∣∣∣∣∣1001x2−x1x22−x2x11x3−x1x32−x3x1∣∣∣∣∣∣=1∗∣B∣对于B=∣∣∣∣x2−x1x22−x2x1x3−x1x32−x3x1∣∣∣∣按列提取公因式则B=(x2−x1)(x3−x1)∣∣∣∣1x21x3∣∣∣∣