# 二次型中标准型所用的特征值的书写顺序有特殊规定吗？没有，但一般按照从小到大，或者从大到小的顺序写——如果有特征向量，则特征值要与特征向量顺序保持一致

## 二、解析

$$A \alpha = \lambda \alpha$$

$$\left[\begin{array}{ccc}a & -1 & 1 \\ -1 & 0 & b \\ 1 & b & -2\end{array}\right]\left[\begin{array}{c}1 \\ -1 \\ 0\end{array}\right]=\lambda\left[\begin{array}{r}1 \\ -1 \\ 0\end{array}\right] \Rightarrow$$

$$\left\{\begin{array}{l}a+1=\lambda \\ -1=-\lambda \\ 1-b=0\end{array} \Rightarrow\left\{\begin{array}{l}\lambda=1 \\ b=1 \\ a=0\end{array}\right.\right.$$

$$A=\left[\begin{array}{ccc}0 & -1 & 1 \\ -1 & 0 & 1 \\ 1 & 1 & -2\end{array}\right] \Rightarrow|\lambda E-A|=0 \Rightarrow$$

$$\left|\begin{array}{ccc}\lambda & 1 & -1 \\ 1 & \lambda & -1 \\ -1 & -1 & \lambda+2\end{array}\right|=0 \Rightarrow$$

$$\left|\begin{array}{ccc}\lambda-1 & 1-\lambda & 0 \\ 0 & \lambda-1 & \lambda+1 \\ -1 & -1 & \lambda+2\end{array}\right|=0 \Rightarrow$$

$$\left|\begin{array}{ccc}\lambda-1 & 2-2 \lambda & 0 \\ 0 & \lambda-1 & \lambda+1 \\ -1 & 0 & \lambda+2\end{array}\right|=0 \Rightarrow$$

$$(\lambda-1)^{2}(\lambda+2)-2(1-\lambda)(\lambda+1) = 0 \Rightarrow$$

$$(\lambda-1)^{2}(\lambda+2)+2(\lambda-1)(\lambda+1)=0 \Rightarrow$$

$$(\lambda-1)[(\lambda-1)(\lambda+2)+2(\lambda+1)]=0 \Rightarrow$$

$$\lambda_{1}=1$$

$$(\lambda-1)(\lambda+2)+2(\lambda+1)=0 \Rightarrow$$

$$\lambda^{2}+\lambda-2+2 \lambda+2=0 \Rightarrow$$

$$\lambda^{2}+3 \lambda=0 \Rightarrow \lambda_{2}=0 \Rightarrow$$

$$\lambda + 3=0 \Rightarrow \lambda_{3}=-3.$$

$$\begin{cases} & \lambda_{1} = 1 \\ & \lambda_{2} = 0 \\ & \lambda_{3} = -3 \end{cases}$$

$$y_{1}^{2} – 3 y_{3}^{2}.$$

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