﻿ 范德蒙德行列式在高等代数解题中的应用

# 范德蒙德行列式在高等代数解题中的应用Application of Vandermonde Determinant in Advanced Algebra Solving

Abstract: The structure and calculation results of Vandermonde determinant are unique, and they are widely used in advanced algebra, calculus and so on. This paper discusses Vandermonde DE determinant in higher algebra problem solving, specifically discussing the application of Vandermonde determinant in polynomial, determinant, linear system of equations, linear correlation of vector group, linear transformation and so on, and selects some of the real exam questions and typical advanced algebra questions to elaborate on these applications.

1. 引言

2. 范德蒙德行列式的定义

${D}_{n}=|\begin{array}{ccccc}1& 1& 1& \cdots & 1\\ {a}_{1}& {a}_{2}& {a}_{3}& \cdots & {a}_{n}\\ {a}_{1}^{2}& {a}_{2}^{2}& {a}_{3}^{2}& \cdots & {a}_{n}^{2}\\ ⋮& ⋮& ⋮& & ⋮\\ {a}_{1}^{n-1}& {a}_{2}^{n-1}& {a}_{3}^{n-1}& \cdots & {a}_{n}^{n-1}\end{array}|$

$n$ 阶行列式称为范德蒙行列式。

${D}_{n}$$n$ 阶范德蒙德行列式 $\left(n\ge 2\right)$ 则有 ${D}_{n}=\underset{1\le i\le j\le n}{\prod }\left({a}_{i}-{a}_{j}\right)$，这里 $\underset{1\le i\le j\le n}{\prod }\left({a}_{i}-{a}_{j}\right)$ 表示所有同类因子 $\left({a}_{i}-{a}_{j}\right)$ (其中 $i>j$ )的乘积：

$\begin{array}{l}\underset{1\le j\le i\le n}{\prod }\left({a}_{i}-{a}_{j}\right)=\left({a}_{n}-{a}_{n-1}\right)\left({a}_{n}-{a}_{n-2}^{}\right)\cdots \left({a}_{n}-{a}_{1}\right)\left({a}_{n-1}-{a}_{n-2}^{}\right)\left({a}_{n-1}-{a}_{n-3}^{}\right)\\ \text{}\cdots \left({a}_{n-1}-{a}_{1}\right)\cdots \left({a}_{3}-{a}_{2}\right)\left({a}_{3}-{a}_{1}\right)\left({a}_{2}-{a}_{1}\right)\end{array}$

3. 范德蒙德行列式的应用

3.1. 范德蒙行列式在多项式中的应用

$\left\{\begin{array}{l}f\left(1\right)+g\left(1\right)+h\left(1\right)=0\hfill \\ f\left(1\right)+{w}_{1}g\left(1\right)+{w}_{{1}^{}}{}^{2}h\left(1\right)=0\hfill \\ f\left(1\right)+{w}_{2}g\left(1\right)+{w}_{2}{}^{2}h\left(1\right)=0\hfill \end{array}$

$\left\{\begin{array}{c}{a}_{0}+{a}_{1}{x}_{1}+\cdots +{a}_{n}{x}_{1}{}^{n}=0\\ {a}_{0}+{a}_{1}{x}_{2}+\cdots +{a}_{n}{x}_{2}{}^{n}=0\\ \cdots \\ {a}_{0}+{a}_{1}{x}_{n}+\cdots +{a}_{n}{x}_{n}{}^{n}=0\end{array}$ (1)

$|\begin{array}{ccccc}1& {x}_{1}& {x}_{1}{}^{2}& \cdots & {x}_{1}{}^{n}\\ 1& {x}_{2}& {x}_{2}{}^{2}& \cdots & {x}_{2}{}^{n}\\ ⋮& ⋮& ⋮& & ⋮\\ 1& {x}_{n}& {x}_{n}{}^{2}& \cdots & {x}_{n}{}^{n}\end{array}|=\underset{1\le i\le j\le n+1}{\overset{}{\prod }}\left({x}_{j}-{x}_{i}\right)\ne 0$

3.2. 范德蒙行列式在行列式中的应用

$A=|\begin{array}{cccc}1& 1& 1& 1\\ 1& 2& 3& 4\\ 1& 4& 9& 16\\ 1& 8& 27& 64\end{array}|$

$|A|=\left(4-3\right)\left(4-2\right)\left(4-1\right)\left(3-2\right)\left(3-1\right)\left(2-1\right)=12$

$|\begin{array}{cccc}1& 1& \cdots & 1\\ {x}_{1}& {x}_{2}& \cdots & {x}_{n}\\ \cdots & \cdots & \cdots & \cdots \\ {x}_{1}^{n-2}& {x}_{2}^{n-2}& \cdots & {x}_{n}^{n-2}\\ {x}_{1}^{n}& {x}_{2}^{n}& \cdots & {x}_{n}^{n}\end{array}|$

$Dn=|\begin{array}{ccccc}1& 1& \cdots & 1& 1\\ {x}_{1}& {x}_{2}& \cdots & {x}_{n}& y\\ \cdots & \cdots & \cdots & \cdots & ⋮\\ {x}_{1}^{n-2}& {x}_{2}^{n-2}& \cdots & {x}_{n}^{n-2}& {y}^{n-2}\\ {x}_{1}^{n-1}& {x}_{2}^{n-1}& \cdots & {x}_{3}^{n-1}& {y}^{{}_{{}^{n-1}}}\\ {x}_{1}^{n}& {x}_{2}^{n}& \cdots & {x}_{n}^{n}& {y}^{n}\end{array}|$

${D}_{n}=\underset{1\le i\le j\le n}{\prod }\left({x}_{i}-{x}_{j}\right)\left(y-{x}_{1}\right)\left(y-{x}_{2}\right)\cdots \left(y-{x}_{n}\right)$

${D}_{n}$ 按最后一列展开，可以得到：

${D}_{n}={p}_{n}{y}^{n}+{p}_{n-1}{y}^{n-1}+\cdots {p}_{1}y+{p}_{0}$

${D}_{n}=\underset{1\le i\le j\le n}{\prod }\left({x}_{i}-{x}_{j}\right)\left(y-{x}_{1}\right)\left(y-{x}_{2}\right)\cdots \left(y-{x}_{n}\right)$

${p}_{n-1}=-\underset{1\le i\le j\le n}{\overset{}{\prod }}\left({x}_{i}-{x}_{j}\right)\cdot \underset{i=1}{\overset{n}{\sum }}{x}_{i}$

$D=-{p}_{n-1}=\underset{1\le i\le j\le n}{\overset{}{\prod }}\left({x}_{i}-{x}_{j}\right)\cdot \underset{i=1}{\overset{n}{\sum }}{x}_{i}$.

3.3. 范德蒙行列式在线性方程组中的应用

$A=\left(\begin{array}{ccc}1& 1& 1\\ 1& 2& a\\ 1& 4& a\end{array}\right),\text{}B=\left(\begin{array}{c}1\\ d\\ {d}^{2}\end{array}\right)$,

A) $a\notin \Omega ,d\notin \Omega$

B) $a\notin \Omega ,d\in \Omega$

C) $a\in \Omega ,d\notin \Omega$

D) $a\in \Omega ,d\in \Omega$

$r\left(A\right)=r\left(A,b\right)<3$,

$r\left(A\right)=r\left(A,b\right)=2$.

$|A|=\left(2-1\right)\left(a-1\right)\left(a-2\right)=\left(a-1\right)\left(a-2\right)$,

$r\left(A\right)=2$,

$|A|=0,\text{}a\in \Omega$

$r\left(A,b\right)=2$,

$|\begin{array}{ccc}1& 1& 1\\ 1& 2& d\\ 1& 4& {d}^{2}\end{array}|=\left(d-1\right)\left(d-2\right)=0$

3.4. 范德蒙行列式在向量组线性相关性中的应用

$\left\{\begin{array}{l}{k}_{0}+{k}_{1}{a}_{1}+{k}_{2}{a}_{1}{}^{2}+\cdots {k}_{N}{a}_{1}{}^{N}=0\hfill \\ {k}_{0}+{k}_{1}{a}_{2}+{k}_{2}{a}_{2}{}^{2}+\cdots {k}_{N}{a}_{2}{}^{N}=0\hfill \\ \text{}\cdots \hfill \\ {k}_{0}+{k}_{1}{a}_{N+1}+{k}_{2}{a}_{N+1}{}^{2}+\cdots {k}_{N}{a}_{N+1}{}^{N}=0\hfill \end{array}$

${a}_{11}{\alpha }_{11}+{a}_{12}{\alpha }_{12}+\cdots {a}_{1{r}_{1}}{\alpha }_{1{r}_{1}}+\cdots {a}_{kr}{}_{{}_{k}}{\alpha }_{k{r}_{k}}=0$,

${\varsigma }_{i}=\underset{j=1}{\overset{ki}{\sum }}{a}_{ij}{\alpha }_{ij}\left(i=1,2,\cdots ,k\right)$

${a}_{11}{\alpha }_{11}+{a}_{12}{\alpha }_{12}+\cdots {a}_{1{r}_{1}}{\alpha }_{1{r}_{1}}+\cdots {a}_{kr}{}_{{}_{k}}{\alpha }_{k{r}_{k}}=0$

${\varsigma }_{1}+{\varsigma }_{2}+\cdots {\varsigma }_{k}=0$,

$A{\varsigma }_{ij}={\lambda }_{i}{\varsigma }_{ij}\left(j=1,2,\cdots ,{r}_{i};\text{}i=1,2,\cdots ,k\right)$,

$A{\varsigma }_{i}=\underset{j=1}{\overset{{r}_{i}}{\sum }}{a}_{ij}A{\alpha }_{ij}=\underset{j=1}{\overset{{r}_{i}^{}}{\sum }}{a}_{ij}{\lambda }_{i}{\alpha }_{ij}={\lambda }_{i}\underset{j=1}{\overset{{r}_{i}}{\sum }}{a}_{ij}{\alpha }_{ij}={\lambda }_{i}{\varsigma }_{i}\left(i=1,2,\cdots ,k\right)$

${\varsigma }_{1}+{\varsigma }_{2}+\cdots {\varsigma }_{k}=0$

$A{\varsigma }_{1}+A{\varsigma }_{2}+\cdots A{\varsigma }_{k}=0$,

$A{\varsigma }_{i}={\lambda }_{i}{\varsigma }_{i}$,

$\lambda {}_{1}\varsigma {}_{1}+{\lambda }_{2}{\varsigma }_{2}+\cdots {\lambda }_{k}{\varsigma }_{k}=0$,

$\lambda {}_{1}{}^{2}{\varsigma }_{1}+{\lambda }_{2}{}^{2}{\varsigma }_{2}+\cdots {\lambda }_{k}{}^{2}{\varsigma }_{k}=0$,

$\left\{\begin{array}{l}{\varsigma }_{1}+{\varsigma }_{2}+\cdots {\varsigma }_{k}=0\hfill \\ {\lambda }_{1}{\varsigma }_{1}+{\lambda }_{2}{\varsigma }_{2}+\cdots {\lambda }_{k}{\varsigma }_{k}=0\hfill \\ {\lambda }_{1}{}^{2}{\varsigma }_{1}+{\lambda }_{2}{}^{2}{\varsigma }_{2}+\cdots {\lambda }_{k}{}^{2}{\varsigma }_{k}=0\hfill \\ \begin{array}{l}\text{}\cdots \\ {\lambda }_{1}{}^{k-1}{\varsigma }_{1}+{\lambda }_{2}{}^{k-1}{\varsigma }_{2}+\cdots {\lambda }_{k}{}^{k-1}{\varsigma }_{k}=0\end{array}\hfill \end{array}$,

$\left({\varsigma }_{1},{\varsigma }_{2},\cdots ,{\varsigma }_{k}\right)|\begin{array}{ccccc}1& {\lambda }_{1}& {\lambda }_{2}& \cdots & {\lambda }_{1}^{m-1}\\ 1& {\lambda }_{2}& {\lambda }_{2}^{2}& \cdots & {\lambda }_{2}^{m-1}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ 1& {\lambda }_{k-1}& {\lambda }_{k-1}^{2}& \cdots & {\lambda }_{k-1}^{k-1}\\ 1& {\lambda }_{k}& {\lambda }_{k}^{2}& \cdots & {\lambda }_{k}^{k-1}\end{array}|=0$

$C=|\begin{array}{ccccc}1& {\lambda }_{1}& {\lambda }_{2}& \cdots & {\lambda }_{1}^{m-1}\\ 1& {\lambda }_{2}& {\lambda }_{2}^{2}& \cdots & {\lambda }_{2}^{m-1}\\ ⋮& ⋮& ⋮& \cdots & ⋮\\ 1& {\lambda }_{k-1}& {\lambda }_{k-1}^{2}& \cdots & {\lambda }_{k-1}^{k-1}\\ 1& {\lambda }_{k}& {\lambda }_{k}^{2}& \cdots & {\lambda }_{k}^{k-1}\end{array}|$

$\left({\varsigma }_{1},{\varsigma }_{2},\cdots ,{\varsigma }_{k}\right)=0$,

${\varsigma }_{i}=0\text{}\left(i=1,2,\cdots ,k\right)$，再将它带入

${\varsigma }_{i}=\underset{j=1}{\overset{ki}{\sum }}{a}_{ij}{\alpha }_{ij}\text{}\left(i=1,2,\cdots ,k\right)$,

$\underset{j=1}{\overset{ki}{\sum }}{a}_{ij}{\alpha }_{ij}=0$,

3.5. 范德蒙行列式在线性变换中的应用

$\alpha =\underset{i=1}{\overset{n}{\sum }}{\alpha }_{i}$,

$T\left(\alpha \right)={k}_{1}T\left({\alpha }_{1}\right)+\cdots {k}_{n}T\left({\alpha }_{n}\right),$

$\left(\alpha ,T\left(\alpha \right),\cdots ,{T}^{n-1}\left(\alpha \right)\right)=\left({\alpha }_{1},\cdots ,{\alpha }_{n}\right)\left(\begin{array}{cccc}{k}_{1}& {k}_{1}{\lambda }_{1}& \cdots & {k}_{1}^{}{\lambda }_{1}^{n-1}{}_{}\\ ⋮& & & ⋮\\ {k}_{n}& {k}_{n}{\lambda }_{n}& \cdots & {k}_{n}{\lambda }_{n}^{n-1}\end{array}\right),$

$r\left(\begin{array}{cccc}{k}_{1}& {k}_{1}{\lambda }_{1}& \cdots & {k}_{1}^{}{\lambda }_{1}^{n-1}{}_{}\\ ⋮& & & ⋮\\ {k}_{n}& {k}_{n}{\lambda }_{n}& \cdots & {k}_{n}{\lambda }_{n}^{n-1}\end{array}\right)=n,$

$\alpha ,T\left(\alpha \right),{T}^{2}\left(\alpha \right),\cdots ,{T}^{n-1}\left(\alpha \right)$ 线性无关，可以看出该式是一个范德蒙德行列式，

$r\left(\begin{array}{cccc}{k}_{1}& {k}_{1}{\lambda }_{1}& \cdots & {k}_{1}^{}{\lambda }_{1}^{n-1}{}_{}\\ ⋮& & & ⋮\\ {k}_{n}& {k}_{n}{\lambda }_{n}& \cdots & {k}_{n}{\lambda }_{n}^{n-1}\end{array}\right)

$\alpha ={k}_{1}{\alpha }_{1}+{k}_{2}{\alpha }_{2}+\cdots +{k}_{i-1}{a}_{i-1}+{k}_{i+1}{a}_{i+1}+\cdots +{k}_{n}{\alpha }_{n}$,

4. 结论

[1] 张凤男. 范德蒙行列式的一些应用[J]. 数学学习与研究, 2018(15): 43.

[2] 张倩. 范德蒙德行列式在线性变换中的应用[J]. 考试周刊, 2017(66): 41.

[3] 侯丽芬. 范德蒙德行列式在行列式计算中的应用[J]. 数学学习与研究, 2020(17): 8-9.

[4] 王萼芳, 石生明. 高等代数[M]. 第5版. 北京: 高等教育出版社, 2019.

[5] 杭州师范大学2014年招收攻读硕士研究生入学《高等代数》考试试题[Z/OL].