“Vandermonde’s determinant”的概念、定义、翻译、参考文献-科学参考

Vandermonde’s determinant

Mathematics

For n real numbers x₁,x₂,…, x_n, the determinant

$| \begin{matrix} \begin{matrix} 1 & x_{1} \\ 1 & x_{2} \end{matrix} & \begin{matrix} x_{1}^{2} & \dots & x_{1}^{n - 1} \\ x_{2}^{2} & \dots & x_{2}^{n - 1} \end{matrix} \\ \begin{matrix} 1 & x_{3} \\ ⋮ & ⋮ \\ 1 & x_{n} \end{matrix} & \begin{matrix} x_{3}^{2} & \dots & x_{3}^{n - 1} \\ ⋮ & \dots & ⋮ \\ x_{n}^{2} & \dots & x_{n}^{n - 1} \end{matrix} \end{matrix} | = \prod_{i > j} (x_{i} - x_{j}) .$

Note that the determinant is non-zero, and so the matrix is invertible, if and only if the x_i are distinct.

单词	Vandermonde’s determinant
释义	Vandermonde’s determinant Mathematics For n real numbers x₁,x₂,…, x_n, the determinant $\| \begin{matrix} \begin{matrix} 1 & x_{1} \\ 1 & x_{2} \end{matrix} & \begin{matrix} x_{1}^{2} & \dots & x_{1}^{n - 1} \\ x_{2}^{2} & \dots & x_{2}^{n - 1} \end{matrix} \\ \begin{matrix} 1 & x_{3} \\ ⋮ & ⋮ \\ 1 & x_{n} \end{matrix} & \begin{matrix} x_{3}^{2} & \dots & x_{3}^{n - 1} \\ ⋮ & \dots & ⋮ \\ x_{n}^{2} & \dots & x_{n}^{n - 1} \end{matrix} \end{matrix} \| = \prod_{i > j} (x_{i} - x_{j}) .$ Note that the determinant is non-zero, and so the matrix is invertible, if and only if the x_i are distinct.
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