PHP开发日志-考研线性代数公式整理

一、普通行列式

（1） $\left|\boldsymbol{A}^{\mathrm{T}}\right|=\left|\boldsymbol{A}\right|$

$\left|\begin{array}{lll}{a_{1}} & {a_{2}} & {a_{3}} \\ {b_{1}} & {b_{2}} & {b_{3}} \\ {c_{1}} & {c_{2}} & {c_{3}}\end{array}\right|=\left|\begin{array}{lll}{a_{1}} & {b_{1}} & {c_{1}} \\ {a_{2}} & {b_{2}} & {c_{2}} \\ {a_{3}} & {b_{3}} & {c_{3}}\end{array}\right|$

（2）上/下三角行列式

$\left|\begin{array}{cccc}{a_{11}} & {a_{12}} & {\cdots} & {a_{1 n}} \\ {} & {a_{22}} & {\cdots} & {a_{2 n}} \\ {} & {} & {\ddots} & {\vdots} \\ {} & {} & {} & {a_{nn}}\end{array}\right|=\left|\begin{array}{cccc}{a_{11}} & {} & {} & {} \\ {a_{21}} & {a_{22}} & {} & {} \\ {\vdots} & {\vdots} & {\ddots} & {} \\ {a_{n 1}} & {a_{n 2}} & {\cdots} & {a_{nn}}\end{array}\right|=a_{11} a_{22} \cdots a_{nn}$

$\left|\begin{array}{ccccc}{a_{11}} & {a_{12}} & {\cdots} & {a_{1 . n-1}} & {a_{1 n}} \\ {a_{21}} & {a_{22}} & {\cdots} & {a_{2,n-1}} & {0} \\ {\vdots} & {\vdots} & {} & {\vdots} & {\vdots} \\ {a_{n 1}} & {0} & {\cdots} & {0} & {0}\end{array}\right|=\left|\begin{array}{cccc}{0} & {\cdots} & {0} & {a_{1 n}} \\ {0} & {\cdots} & {a_{2,n-1}} & {a_{2 n}} \\ {\vdots} & {} & {\vdots} & {\vdots} \\ {a_{n 1}} & {\cdots} & {a_{n, n-1}} & {a_{nn}}\end{array}\right|=(-1)^{\frac{n(n-1)}{2}} a_{1 n} a_{2, n-1} \cdots a_{n 1}$

（3）范德蒙行列式

$\left|\begin{array}{}{1} & {1} & {\dots} & {1} \\ {x_{1}} & {x_{2}} & {\dots} & {x_{n}} \\ {\vdots} & {\vdots} & {} & {\vdots} \\ {x_{1}^{n-1}} & {x_{2}^{n-1}} & {\dots} & {x_{n}^{n-1}}\end{array}\right|$

$\begin{array}{c}{\prod_{1 \leq j<i \leq n}\left(x_{i}-x_{j}\right)=\left(x_{n}-x_{n-1}\right)\left(x_{n}-x_{n-2}\right) \cdots\left(x_{n}-x_{1}\right)\left(x_{n-1}-x_{n-2}\right)\left(x_{n-1}-x_{n-3}\right) \cdots} \\ {\left(x_{n-1}-x_{1}\right) \cdots\left(x_{3}-x_{2}\right)\left(x_{3}-x_{1}\right)\left(x_{2}-x_{1}\right)}\end{array}$

例：

$D=\left|\begin{array}{}{1} & {1} & {1} & {1} \\ {2} & {2^{2}} & {2^{3}} & {2^{4}} \\ {3} & {3^{2}} & {3^{3}} & {3^{4}} \\ {4} & {4^{2}} & {4^{3}} & {4^{4}}\end{array}\right|=2 \cdot 3 \cdot 4\left|\begin{array}{}{1} & {1} & {1} & {1} \\ {1} & {2} & {2^{2}} & {2^{3}} \\ {1} & {3} & {3^{2}} & {3^{3}} \\ {1} & {4} & {4^{2}} & {4^{3}}\end{array}\right|=4 !\left|\begin{array}{}{1} & {1} & {1} & {1} \\ {1} & {2} & {3} & {4} \\ {1} & {2^{2}} & {3^{2}} & {4^{2}} \\ {1} & {2^{3}} & {3^{3}} & {4^{3}}\end{array}\right| \\=4 !(4-3)(4-2)(4-1)(3-2)(3-1)(2-1)=4 ! 3 ! 2 !=\prod_{i=1}^{4}k!$

这道题需要注意的是，由于公式（1），导致范德蒙德除了定义中的情况，在某些情况下还可以用于其转置矩阵。

（4）拉普拉斯展开式特殊形式

$\left|\begin{array}{ll}{\boldsymbol{A}} & {\boldsymbol{O}} \\ {\boldsymbol{O}} & {\boldsymbol{B}}\end{array}\right|=\left|\begin{array}{ll}{\boldsymbol{A}} & {\boldsymbol{O}} \\ {\boldsymbol{C}} & {\boldsymbol{B}}\end{array}\right|=\left|\begin{array}{ll}{\boldsymbol{A}} & {\boldsymbol{C}} \\ {\boldsymbol{O}} & {\boldsymbol{B}}\end{array}\right|=|\boldsymbol{A}||\boldsymbol{B}|$

$\left|\begin{array}{cc}{\boldsymbol{O}} & {\boldsymbol{A}} \\ {\boldsymbol{B}} & {\boldsymbol{O}}\end{array}\right|=\left|\begin{array}{cc}{\boldsymbol{O}} & {\boldsymbol{A}} \\ {\boldsymbol{B}} & {\boldsymbol{C}}\end{array}\right|=\left|\begin{array}{cc}{\boldsymbol{C}} & {\boldsymbol{A}} \\ {\boldsymbol{B}} & {\boldsymbol{O}}\end{array}\right|=(-1)^{mn}|\boldsymbol{A}||\boldsymbol{B}|$

注意这些都是方阵

例题： $D=\left|\begin{array}{llll}{1} & {0} & {0} & {4} \\ {0} & {2} & {3} & {0} \\ {0} & {3} & {2} & {0} \\ {4} & {0} & {0} & {1}\end{array}\right|=-\left|\begin{array}{cccc}{1} & {0} & {0} & {4} \\ {4} & {0} & {0} & {1} \\ {0} & {3} & {2} & {0} \\ {0} & {2} & {3} & {0}\end{array}\right|=\left|\begin{array}{cccc}{1} & {4} & {0} & {0} \\ {4} & {1} & {0} & {0} \\ {0} & {0} & {2} & {3} \\ {0} & {0} & {3} & {2}\end{array}\right|=\left|\begin{array}{cc}{1} & {4} \\ {4} & {1}\end{array}\right| \cdot\left|\begin{array}{cc}{2} & {3} \\ {3} & {2}\end{array}\right|=75$

$\begin{align}D&=\left|\begin{array}{cccc}{a_{1}} & {0} & {a_{2}} & {0} \\ {0} & {b_{1}} & {0} & {b_{2}} \\ {c_{1}} & {0} & {c_{2}} & {0} \\ {0} & {d_{1}} & {0} & {d_{2}}\end{array}\right|=-\left|\begin{array}{cccc}{a_{1}} & {0} & {a_{2}} & {0} \\ {c_{1}} & {0} & {c_{2}} & {0} \\ {0} & {b_{1}} & {0} & {b_{2}} \\ {0} & {d_{1}} & {0} & {d_{2}}\end{array}\right|=\left|\begin{array}{llll}{a_{1}} & {a_{2}} & {0} & {0} \\ {c_{1}} & {c_{2}} & {0} & {0} \\ {0} & {0} & {b_{1}} & {b_{2}} \\ {0} & {0} & {d_{1}} & {d_{2}}\end{array}\right| \\&\begin{array}{l}{=\left|\begin{array}{ll}{a_{1}} & {a_{2}} \\ {c_{1}} & {c_{2}}\end{array}\right| \cdot\left|\begin{array}{ll}{b_{1}} & {b_{2}} \\ {d_{1}} & {d_{2}}\end{array}\right|} = {\left(a_{1} c_{2}-a_{2} c_{1}\right)\left(b_{1} d_{2}-b_{2} d_{1}\right)}\end{array}\end{align}$

(4)若 $\boldsymbol{A}=\left[\begin{array}{llll}{0} & {a} & {b} & {c} \\ {0} & {0} & {d} & {e} \\ {0} & {0} & {0} & {f} \\ {0} & {0} & {0} & {0}\end{array}\right]$ ,则 $A^{3}=\left[\begin{array}{cccc}{0} & {0} & {0} & {a d f} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0} \\ {0} & {0} & {0} & {0}\end{array}\right], \mathbf{A}^{4}=\mathbf{O}$

二、方阵行列式

1. $\left|\boldsymbol{A}^{\mathrm{T}}\right|=|\boldsymbol{A}|$

2.若 $AB=E$ 则 $AB=BA$ 且 $A$ 和 $B$ 均满秩

3. $|k A|=k^{n}|A|$

4. $|\boldsymbol{A} \boldsymbol{B}|=|\boldsymbol{A}||\boldsymbol{B}|\ \ \ \left|\boldsymbol{A}^{2}\right|=|\boldsymbol{A}|^{2}$

5. $\left|A^{*}\right|=|A|^{n-1}$

6. $\left|A^{-1}\right|=|A|^{-1}$

7. $|A|=\prod_{i=1}^{n} \lambda_{i} \ \ \ \ \ \ \ \lambda_{i}(i=1,2, \cdots, n)$ 是特征值

8. $\boldsymbol{A} \sim \boldsymbol{B}$ 则 $|\boldsymbol{A}|=|\boldsymbol{B}|$ ， $tr(A)=tr(B)$

三、矩阵运算

1. ${A A^{*}=A^{*} A=|A| E}$

2. ${\left(A^{*}\right)^{-1}=\left(A^{-1}\right)^{*}=\frac{1}{|A|} A(|A| \neq 0)}$

3. ${\left(A^{*}\right)^{T}=\left(A^{T}\right)^{*}}$

4. ${(k A)^{*}=k^{n-1} A^{*}}$

5. $\left(A^{*}\right)^{*}=|A|^{n-2} A \quad(n \geqslant 2)$

6. $(k A)^{\mathrm{T}}=k A^{\mathrm{T}}$

7. $(A B)^{\mathrm{T}}=B^{\mathrm{T}} A^{\mathrm{T}}$

8. $(k A)^{-1}=\frac{1}{k} A^{-1}$

9. $(\boldsymbol{A B})^{-1}=\boldsymbol{B}^{-1} \mathbf{A}^{-1}$

10.如果 $\boldsymbol{A}^{\mathrm{T}}$ 可逆, $\left(\boldsymbol{A}^{\mathrm{T}}\right)^{-1}=\left(\boldsymbol{A}^{-1}\right)^{\mathrm{T}}$

11. 若 $|A|\ne0$ 则 $A^{-1}=\frac{1}{|A|} A^{*}$

12.如果 $r(A)=1$ ， $则有l=\sum{a_{ii}},A^m=l^{m-1}A$

13.初等矩阵变换公式 $E_{i}^{-1}(k)=E_{i}\left(\frac{1}{k}\right), E_{i j}^{-1}=E_{i j}, E_{i j}^{-1}(k)=E_{i j}(-k)$

14 矩阵 $A$ 和矩阵 $B$ 等价 $\Leftrightarrow$ $R(A)=R(B)$

15.如果 $A$ 可逆， $r(AB)=r(B),r(BA)=r(B)$

16. $r\left(\boldsymbol{A}^{*}\right)=\left\{\begin{array}{ll}{n,} & {r(\boldsymbol{A})=n} \\ {1,} & {r(\boldsymbol{A})=n-1} \\ {0,} & {r(\boldsymbol{A})<n-1}\end{array}\right.$

17.分块矩阵求逆 $\left[\begin{array}{ll}{\boldsymbol{B}} & {\boldsymbol{O}} \\ {\boldsymbol{O}} & {\boldsymbol{C}}\end{array}\right]^{-1}=\left[\begin{array}{ll}{\boldsymbol{B}^{-1}} & {\boldsymbol{O}} \\ {\boldsymbol{O}} & {\boldsymbol{C}^{-1}}\end{array}\right] ;\left[\begin{array}{ll}{\boldsymbol{O}} & {\boldsymbol{B}} \\ {\boldsymbol{C}} & {\boldsymbol{O}}\end{array}\right]^{-1}=\left[\begin{array}{cc}{\boldsymbol{O}} & {\boldsymbol{C}^{-1}} \\ {\boldsymbol{B}^{-1}} & {\boldsymbol{O}}\end{array}\right]$

四、矩阵与向量

1. $A B=A\left[\beta_{1}, \beta_{2}, \cdots, \beta_{s}\right]=\left[A \beta_{1}, A \beta_{2}, \cdots, A \beta_s\right]$

2. $\left[\begin{array}{cccc}{a_{11}} & {a_{12}} & {\cdots} & {a_{1 n}} \\ {a_{21}} & {a_{22}} & {\cdots} & {a_{2 n}} \\ {\vdots} & {\vdots} & {} & {\vdots} \\ {a_{m 1}} & {a_{m 2}} & {\cdots} & {a_{m n}}\end{array}\right]\left[\begin{array}{c}{\boldsymbol{\beta}_{1}} \\ {\boldsymbol{\beta}_{2}} \\ {\vdots} \\ {\boldsymbol{\beta}_{n}}\end{array}\right]=\left[\begin{array}{c}{\boldsymbol{\alpha}_{1}} \\ {\boldsymbol{\alpha}_{2}} \\ {\vdots} \\ {\boldsymbol{\alpha}_{n}}\end{array}\right]$ 可以得到：

$\left\{\begin{array}{c}{a_{11} \boldsymbol{\beta}_{1}+a_{12} \boldsymbol{\beta}_{2}+\cdots+a_{1 n} \boldsymbol{\beta}_{n}=\boldsymbol{\alpha}_{1}} \\ {a_{21} \boldsymbol{\beta}_{1}+a_{22} \boldsymbol{\beta}_{2}+\cdots+a_{2 n} \boldsymbol{\beta}_{n}=\boldsymbol{\alpha}_{2}} \\ {\vdots} \\ {a_{m 1} \boldsymbol{\beta}_{1}+a_{m 2} \boldsymbol{\beta}_{2}+\cdots+a_{m n} \boldsymbol{\beta}_{n}=\boldsymbol{\alpha}_{m}}\end{array}\right.$

3. $\left[\gamma_{1}, \gamma_{2}, \cdots, \gamma_{n}\right]\left[\begin{array}{cccc}{b_{11}} & {b_{12}} & {\cdots} & {b_{1 s}} \\ {b_{21}} & {b_{22}} & {\cdots} & {b_{2 s}} \\ {\vdots} & {\vdots} & {} & {\vdots} \\ {b_{n 1}} & {b_{n 2}} & {\cdots} & {b_{n s}}\end{array}\right]=\left[\begin{array}{c}{\boldsymbol{\delta}_{1}, \boldsymbol{\delta}_{2}, \cdots, \boldsymbol{\delta}_{s}} \end{array}\right]$

$\left\{\begin{array}{c}{b_{11} \gamma_{1}+b_{21} \gamma_{2}+\cdots+b_{n 1} \gamma_{n}=\delta_{1}} \\ {b_{12} \gamma_{1}+b_{22} \gamma_{2}+\cdots+b_{n 2} \gamma_{n}=\delta_{2}} \\ {\vdots} \\ {b_{1 s} \gamma_{1}+b_{2 s} \gamma_{2}+\cdots+b_{ns} \gamma_{s}=\delta_{s}}\end{array}\right.$

4.内积 $(\boldsymbol{\alpha}, \boldsymbol{\beta})=a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}=\boldsymbol{\alpha}^{\mathrm{T}} \boldsymbol{\beta}=\boldsymbol{\beta}^{\mathrm{T}} \boldsymbol{\alpha}$

五、齐次方程组

1.齐次方程组 $A_{n\times s}=0$ 的基础解系的个数是解空间的维数,为 $n-r(A)$

2.齐次方程组 $A_{n\times s}x=0$ 有非零解 $\Leftrightarrow r(A)<s\Leftrightarrow a_1,a_2...a_s$ 线性相关

齐次方程组 $A_{n\times s}x=0$ 只有零解 $\Leftrightarrow r(A)=s\Leftrightarrow a_1,a_2...a_s$ 线性无关

六、非齐次方程组

1.若非齐次方程组 $Ax=b$ 有特解 $\boldsymbol{\eta}_{1}、\boldsymbol{\eta}_{2}$ ，则 $\frac12(\boldsymbol{\eta}_{1}+\boldsymbol{\eta}_{2})$ 也是该非齐次方程的特解，而 $\boldsymbol{\eta}_{1}-\boldsymbol{\eta}_{2}$ 则为其对应齐次方程的基础解系。

七、特征值

1.秩为1的矩阵的特征值为 $\sum{a_{ii}}$ ,0,0

2.若 $\alpha、\beta$ 都是 $n$ 维列向量，则 $\alpha\beta^T$ 与 $\beta\alpha^T$ 都是秩为1的矩阵，且 $\alpha^T\beta=\beta^T\alpha=tr(\alpha\beta^T)=tr(\beta\alpha^T)$ ，故其特征值为 $\alpha\beta^T$ 或 $\beta\alpha^T$

3. $\sum_{i=1}^{n} \lambda_{i}=\sum_{i=1}^{n} a_{i i}=tr(A)$

4. $\prod_{i=1}^{n} \lambda_{i}=|\boldsymbol{A}|$

5.对角阵 $\Lambda=\left[\begin{array}{lll}{a} & {0} & {0} \\ {0} & {b} & {0} \\ {0} & {0} & {c}\end{array}\right]$ 、上下三角阵的特征值即是主对角元素

6.若 $A\alpha=\lambda\alpha$ 则有 $A^{*} \alpha=\frac{|A|}{\lambda} \alpha$ ，所以 $A^*$ 特征值是 $\frac{|A|}{\lambda}$

7. $\begin{array}{|c|c|c|c|c|}\hline 矩阵 & A & {k A} & {A^{k}} & {f(A)} & {A^{-1}} & {A}&A^{-1}+f(A) \\ \hline 特征值&\lambda & {k \lambda} & {\lambda^{*}} & {f(\lambda)} & {\lambda^{-1}} & {\frac{|A|}{\lambda}} &\frac1\lambda+f(\lambda)\\ \hline 对应特征向量&a & {a} & {a} & {a} & {a} & {a}& {a}\\ \hline\end{array}$

注意其中 $f$ 是多项式.

8.实对称矩阵的不同特征值对应的特征向量两两正交。

9.两个实对称矩阵若有相同特征值，则其必相似。

10.若 $\alpha1$ 和 $\alpha2$ 都是 $A$ 属于特征值 $\lambda_0$ 的特征向量，则 $k_1\alpha_1+k_2\alpha_2$ 也是 $A$ 属于特征值 $\lambda_0$ 的特征向量。

11.由惯性定理可知，对于一个二次型其特征值正值的个数等于其正惯性指数，其特征值负值的个数等于其负惯性指数，而知道其正负惯性指数则可以直接写出其规范型。

八、题型

1.先使用矩阵运算公式，后使用行列式公式

已知 $A$ 是3阶矩阵， $A^T$ 是 $A$ 的转置矩阵， $A^*$ 是 $A$ 的伴随矩阵，如果 $|A|=\frac14$ ,则 $\left|\left(\frac{2}{3} A\right)^{-1}-8 A^{*}\right|=$ ____________

$\left|\left(\frac{2}{3} A\right)^{-1}-8 A^{*}\right|=\left|\frac{3}{2} A^{-1}-2 A^{-1}\right|=\left|-\frac{1}{2} A^{-1}\right|=\left(-\frac{1}{2}\right)^{3}\left|A^{-1}\right|=-\frac{1}{2}$

2.利用初等变换矩阵进行计算

已知 $A=\left[\begin{array}{}{a_{11}} & {a_{12}} & {a_{13}} \\ {a_{21}} & {a_{22}} & {a_{23}} \\ {a_{31}} & {a_{32}} & {a_{33}}\end{array}\right], B=\left[\begin{array}{}{a_{11}-2 a_{13}} & {a_{12}} & {a_{12}+a_{13}} \\ {a_{21}-2 a_{23}} & {a_{22}} & {a_{22}+a_{23}} \\ {a_{31}-2 a_{33}} & {a_{32}} & {a_{32}+a_{33}}\end{array}\right]$ ,若行列式 $|A|=2$ ,则 $\left(\boldsymbol{A}^{*} \boldsymbol{B}\right)^{-1}=$ ______

$\boldsymbol{B}=\boldsymbol{A}\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {-2} & {0} & {1}\end{array}\right]\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {1} \\ {0} & {0} & {1}\end{array}\right]$

$\begin{array}{lll}{A^{*} B=A^{*} A} {\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {-2} & {0} & {1}\end{array}\right]\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {1} \\ {0} & {0} & {1}\end{array}\right]=2\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {-2} & {0} & {1}\end{array}\right]\left[\begin{array}{lll}{1} & {0} & {0} \\ {0} & {1} & {1} \\ {0} & {0} & {1}\end{array}\right]}\end{array}$

$\begin{align}\left(\boldsymbol{A}^{*} \boldsymbol{B}\right)^{-1}&=\frac{1}{2}\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {1} \\ {0} & {0} & {1}\end{array}\right]^{-1}\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {0} \\ {-2} & {0} & {1}\end{array}\right]^{-1}=\frac{1}{2}\left[\begin{array}{ccc}{1} & {0} & {0} \\ {0} & {1} & {-1} \\ {0} & {0} & {1}\end{array}\right]\left[\begin{array}{cc}{1} & {0}& {0} \\ {0} & {1} & {0}\\ {2} & {0}& {1}\end{array}\right]\\&=\frac{1}{2}\left[\begin{array}{ccc}{1} & {0} & {0} \\ {-2} & {1} & {-1} \\ {2} & {0} & {1}\end{array}\right]\end{align}$

3.利用矩阵求解齐次方程组

若 $AB=0$ ，则 $B$ 的每一个列向量都是 $Ax=0$ 的解

例题：已知 $A=\left[\begin{array}{ccc}{1} & {-2} & {0} \\ {2} & {1} & {5} \\ {0} & {1} & {1}\end{array}\right]$ ， $A^*$ 是 $A$ 的伴随矩阵，则 $A^*x=0$ 的通解是

解：

经计算， $\left| A \right|=0$ ， $r(A)=2$ 那么 $r(A^*)=1$ ,从而 $n-r\left(A^{*}\right)=2$ ， $A^*x=0$ 有两个基础解系，通解形式为 $k_{1} \boldsymbol{\eta}_{1}+k_{2} \boldsymbol{\eta}_{2}$ ，而 $A^{*} A=|A| E=O$

所以 $A$ 的列向量是 $A^*x=0$ 的解

4.利用四则运算求特征值

例题：已知 $A=\left[\begin{array}{lll}{3} & {2} & {2} \\ {2} & {3} & {2} \\ {2} & {2} & {3}\end{array}\right]$ 求 $A$ 的特征值

解： $A=\left[\begin{array}{ll}{1} & {}& {} \\ {} & {1} \\ {} & & {} {1}\end{array}\right]+\left[\begin{array}{lll}{2} & {2} & {2} \\ {2} & {2} & {2} \\ {2} & {2} & {2}\end{array}\right]=E+B$

由 $r(B)=1$ 知 $B$ 的特征值是 $\sum_{i=1}^{3} a_{i i}=6,0,0$ 故 $A$ 的特征值是 7，1，1

5.行列式加法转化为乘法

例题：

已知 $A$ 是3阶矩阵，特征值为1，2，-2,则 $|A+A^{-1}|=$

解： $A+A^{-1}=A^{-1}\left(A^{2}+E\right)$

由于 $A$ 的特征值为1,2,-2故 $A^2+E$ 的特征值为2，5，5

又 $|\boldsymbol{A}|=1 \cdot 2 \cdot(-2)=-4$

故 $\left|\boldsymbol{A}+\boldsymbol{A}^{-1}\right|=-\frac{25}{2}$

6.向量得到矩阵公式

例题：设 $A$ 是三阶矩阵， $\alpha1、\alpha2、\alpha3$ 是三维线性无关的列向量，且 $A\alpha1=\alpha2+\alpha3,A\alpha2=\alpha1+\alpha3,A\alpha3=\alpha1+\alpha2$ 则和 $A$ 相似的矩阵是_____

根据已知条件有 $A\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)=\left(\alpha_{2}+\alpha_{3}, \alpha_{1}+\alpha_{3}, \alpha_{1}+\alpha_{2}\right)=\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right)\left[\begin{array}{ccc}{0} & {1} & {1} \\ {1} & {0} & {1} \\ {1} & {1} & {0}\end{array}\right]$ 记 $P=\left(\alpha_{1}, \alpha_{2}, \alpha_{3}\right), B=\left[\begin{array}{lll}{0} & {1} & {1} \\ {1} & {0} & {1} \\ {1} & {1} & {0}\end{array}\right]$

故 $A P=P B \Rightarrow P^{-1} A P=B$

故 $A$ 的相似矩阵是 $\left[\begin{array}{lll}{0} & {1} & {1} \\ {1} & {0} & {1} \\ {1} & {1} & {0}\end{array}\right]$

7.求特征值，求秩转化为求行列式

例：

已知二次型 $x^{\mathrm{T}} A x=a x_{1}^{2}+2 x_{2}^{2}+a x_{3}^{2}+6 x_{1} x_{2}+2 x_{2} x_{3}$ 的秩为2，则a=____

解： $A=\left[\begin{array}{lll}{a} & {3} & {0} \\ {3} & {2} & {1} \\ {0} & {1} & {a}\end{array}\right]$ ，由于秩为2，则 $|A|=0$ 解得a=0或5

例:

二次型 $\boldsymbol{x}^{\top} \boldsymbol{A} \boldsymbol{x}=a x_{1}^{2}+a x_{2}^{2}+2 x_{3}^{2}-2 x_{1} x_{2}+2 x_{1} x_{3}-2 x_{2} x_{3}$ 经过正交变换化为标准型 $3 y_{1}^{2}-y_{3}^{2}$ 则a=_____