2.1
基础循环矩阵
1. 设n阶基础循环矩阵
$$ A=\left (\begin{array}{llll} 0 &1 & 0 &\cdots & 0 \\ 0 &0 & 1 &\cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 &0 &… & 1 \\ 1 & 0 &0 &… & 0 \\ \end{array}\right) $$ 求证: $$ A^{k}=\left (\begin{array}{llll} O &I_{n-k}\\ I_{k} &O\\ \end{array}\right),1{\le}k{\le}n $$
幂零Jordan块
2. 设n阶幂零Jordan块
$$ A=\left (\begin{array}{llll} 0 &1 & 0 &\cdots & 0 \\ 0 &0 & 1 &\cdots & 0 \\ \vdots & \vdots & \vdots & & \vdots \\ 0 & 0 &0 &… & 1 \\ 0 & 0 &0 &… & 0 \\ \end{array}\right) $$ 求证: $$ A^{k}=\left (\begin{array}{llll} O &I_{n-k}\\ O &O\\ \end{array}\right) $$
多项式的友阵与Frobenius块
3. 设首一多项式
$f(x)=x^{n}+a_{1}x^{n-1}+\cdots+a_{n-1}x+a_{n}$, $f(x)$的友阵
$$ C(f(x))=\left (\begin{array}{rrrr} 0 &0 &\cdots 0 & -a_{n} \\ 1 &0 &\cdots 0 & -a_{n-1} \\ 0 &1 &\cdots 0 & -a_{n-2} \\ \vdots &\vdots \ &\vdots & \vdots \\ 0 & 0 &\cdots 1 & -a_{1}\\ \end{array}\right) $$ 求证:$\mid xI_{n}-C(f(x)) \mid=f(x)$,$C(f(x))$的转置$F(f(x))$称作$f(x)$的Frobenius块
对称阵与反对称
4. 设$A$为n阶对称阵,求证:$A$是零矩阵的充要条件是对任意的n维列向量$α$,有$$α’Aα=0$$
(根据以上结论可以证明下列基本结论,反对称矩阵的刻画条件)
- 设$A$为n阶方阵,求证:$A$是反对称矩阵的充要条件是对任意的n维列向量$α$,有 $$α’Aα=0$$