LA-5-特征值与特征向量
2021-08-14 16:47:39 0 举报AI智能生成
线性代数 第五章 特征值与特征向量 知识点梳理
线性代数
特征值与特征向量
相似矩阵
实对称矩阵
考研
大学教育
模版推荐
作者其他创作
大纲/内容
相似矩阵
定义
若<span class="equation-text" data-index="0" data-equation="\exists " contenteditable="false"><span></span><span></span></span>可逆矩阵P, P<span class="equation-text" data-index="1" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP=B，称A<span class="equation-text" data-index="2" data-equation="\sim " contenteditable="false"><span></span><span></span></span>B
若A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation=" \Lambda " contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="2" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>为对角矩阵，称A可相似对角化，<br><span class="equation-text" data-index="3" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>是A的相似标准型<br>
性质
反身性：A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span>A
P=E
对称性：若A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span>B，则B<span class="equation-text" data-index="1" data-equation="\sim" contenteditable="false"><span></span><span></span></span>A
P<span class="equation-text" data-index="0" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP=B，PBP<span class="equation-text" data-index="1" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>=A
传递性：若A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span>B，B<span class="equation-text" data-index="1" data-equation="\sim" contenteditable="false"><span></span><span></span></span>C，则A<span class="equation-text" data-index="2" data-equation="\sim" contenteditable="false"><span></span><span></span></span>C
P<span class="equation-text" data-index="0" data-equation="_1^{-1}" contenteditable="false"><span></span><span></span></span>AP<span class="equation-text" data-index="1" data-equation="_1" contenteditable="false"><span></span><span></span></span>=B，P<span class="equation-text" data-index="2" data-equation="_2^{-1}" contenteditable="false"><span></span><span></span></span>BP<span class="equation-text" data-index="3" data-equation="_2" contenteditable="false"><span></span><span></span></span>=C，<br>(P<span class="equation-text" data-index="4" data-equation="_2^{-1}" contenteditable="false"><span></span><span></span></span>P<span class="equation-text" data-index="5" data-equation="_1^{-1}" contenteditable="false"><span></span><span></span></span>)A(P<span class="equation-text" data-index="6" data-equation="_1" contenteditable="false"><span></span><span></span></span>P<span class="equation-text" data-index="7" data-equation="_2" contenteditable="false"><span></span><span></span></span>)=P<span class="equation-text" data-index="8" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP=C<br>
运算不变性<br>当A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span>B<br>
A<span class="equation-text" data-index="0" data-equation="^n" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="\sim" contenteditable="false"><span></span><span></span></span>B<span class="equation-text" data-index="2" data-equation="^n" contenteditable="false"><span></span><span></span></span>
A+kE<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span>B+kE
if A可逆，则A<span class="equation-text" data-index="0" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="\sim" contenteditable="false"><span></span><span></span></span>B<span class="equation-text" data-index="2" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span><br>
推论<br>A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span>B时<br>
特征多项式相同，ie.|<span class="equation-text" data-index="0" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>E-A|=|<span class="equation-text" data-index="1" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>E-B|
特征值相同，ie.<span class="equation-text" data-index="0" data-equation="\lambda_A=\lambda_B" contenteditable="false"><span></span><span></span></span>
|A|=|B|=<span class="equation-text" data-index="0" data-equation="\prod\lambda" contenteditable="false"><span></span><span></span></span>
|B|=|P<span class="equation-text" data-index="0" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP|=|P<span class="equation-text" data-index="1" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>||A||P|
<span class="equation-text" data-index="0" data-equation="\sum a_{ii}=\sum b_{ii}=\sum\lambda" contenteditable="false"><span></span><span></span></span>
r(A)=r(B)
r(B)=r(P<span class="equation-text" data-index="0" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP)=r(AP)=r(A)
<font color="#F44336">相似对角化<br>A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>??<br></font>
定义：A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>，称A可以相似对角化<br>
<font color="#F44336">结论：即P就是A的3个特征向量，对角矩阵的对角线就是3个特征值<br></font>
没有n个线性无关的特征向量的矩阵B<span class="equation-text" data-index="0" data-equation="\nsim" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>
<font color="#F44336">充分条件</font>：A有n个<font color="#F44336">不同的</font>特征值
<font color="#F44336">充要条件</font>：A有n个<font color="#F44336">线性无关</font>特征向量<br>
证明<br>
将特征向量拼成P<span class="equation-text" data-index="0" data-equation="_{3\times3}" contenteditable="false"><span></span><span></span></span>，由于线性无关，P可逆。<br><span class="equation-text" data-index="1" data-equation="AP=(A\alpha_1,A\alpha_2,A\alpha_3)=(\lambda_1\alpha_1,\lambda_2\alpha_2,\lambda_3\alpha_3)" contenteditable="false"><span></span><span></span></span><br><span class="equation-text" data-index="2" data-equation="=(\alpha_1,\alpha_2,\alpha_3)\begin{bmatrix}\lambda_1&&\\&\lambda_2&\\&&\lambda_3\end{bmatrix}=P\Lambda" contenteditable="false"><span></span><span></span></span><br>ie. AP=P<span class="equation-text" data-index="3" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>，整理得P<span class="equation-text" data-index="4" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP=<span class="equation-text" data-index="5" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>，即A<span class="equation-text" data-index="6" data-equation="\sim" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="7" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span><br>
已知A<span class="equation-text" data-index="0" data-equation="\sim" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>，P<span class="equation-text" data-index="2" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP=<span class="equation-text" data-index="3" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>，有AP=P<span class="equation-text" data-index="4" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span><br>将P列分块为(<span class="equation-text" data-index="5" data-equation="\alpha_1,\alpha_2,\alpha_3" contenteditable="false"><span></span><span></span></span>)，显然有<br><span class="equation-text" data-index="6" data-equation="(A\alpha_1,A\alpha_2,A\alpha_3)=(\alpha_1,\alpha_2,\alpha_3)\begin{bmatrix}a&&\\&b&\\&&c\end{bmatrix}\\=(a\alpha_1,b\alpha_2,c\alpha_3)" contenteditable="false"><span></span><span></span></span><br>由于P可逆，必有<span class="equation-text" data-index="7" data-equation="\alpha_1,\alpha_2,\alpha_3" contenteditable="false"><span></span><span></span></span>线性无关，即A有<br>3个特征向量线性无关<br>
<font color="#F44336">充要条件2</font>：若A<span class="equation-text" data-index="0" data-equation="\sim\Lambda" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="1" data-equation="\Leftrightarrow" contenteditable="false"><span></span><span></span></span><span class="equation-text" data-index="2" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>是A的<font color="#F44336">k重</font>特征值，则<span class="equation-text" data-index="3" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>必有<font color="#F44336">k个线性无关</font>的特征向量
考察
已知待定矩阵相似，求未知数
构造方程
性质：
特征多项式相同
<span class="equation-text" data-index="0" data-equation="\sum\lambda_i" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\sum a_{ii}" contenteditable="false"><span></span><span></span></span><br>
|A|=<span class="equation-text" data-index="0" data-equation="\prod\lambda_i" contenteditable="false"><span></span><span></span></span><br>
运算不变性
结论：对角矩阵是A的特征值
例：已知<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}2&0&0\\0&0&1\\0&1&x\end{bmatrix}" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="1" data-equation="\begin{bmatrix}2&0&0\\0&y&0\\0&0&-1\end{bmatrix}" contenteditable="false"><span></span><span></span></span>相似<br>1.求x,y；2.求可逆矩阵P使P<span class="equation-text" data-index="2" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP=B。<br>
把给定矩阵相似对角化
求对角矩阵
求特征值
求可逆矩阵P
求特征向量<br>
构造可逆矩阵P=(<span class="equation-text" data-index="0" data-equation="\alpha_1,\alpha_2" contenteditable="false"><span></span><span></span></span>...)
写出P<span class="equation-text" data-index="0" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP=<span class="equation-text" data-index="1" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>
注意细节：特征值和<br>特征向量位置要匹配<br>
证明给定矩阵是否可以相似对角化<br>
使用充要条件2
实对称矩阵<br>
定义
即元素均为实数的对称矩阵<br>
定理
<font color="#F44336">必定可以相似对角化</font>
无论根的情况
不同特征值对应的特征向量相互正交
Cmp：<font color="#F44336">一般矩阵</font>不同特征值对应的特征向量<font color="#F44336">线性无关</font>
存在正交矩阵，可用来相似对角化<br>
Cmp：<font color="#F44336">一般矩阵</font>只能用<font color="#F44336">可逆矩阵</font>相似对角化
考察<br>
定理应用
实对称矩阵A<span class="equation-text" data-index="0" data-equation="_{3\times3}" contenteditable="false"><span></span><span></span></span>满足A<span class="equation-text" data-index="1" data-equation="^2" contenteditable="false"><span></span><span></span></span>=A，且r(A-E)=2，求A的特征值<br>
把给定矩阵相似对角化
求对角矩阵
求特征值
求可逆矩阵P
若要求正交矩阵P，特征向量必<br>须检验，满足相互正交且单位化<br>
特征向量两两正交→单位化<br>
特征向量不正交→施密特正交单位化
令<span class="equation-text" data-index="0" data-equation="b_k=a_k-\sum\limits_{i=1}^{k-1}\frac{[b_i,a_k]}{[b_i,b_i]}b_i" contenteditable="false"><span></span><span></span></span><br>再令<span class="equation-text" data-index="1" data-equation="e_k=\hat{b_k}" contenteditable="false"><span></span><span></span></span><br>
尤其注意[b<span class="equation-text" data-index="0" data-equation="_i" contenteditable="false"><span></span><span></span></span>,a<span class="equation-text" data-index="1" data-equation="_k" contenteditable="false"><span></span><span></span></span>]为0时可以忽略此项。<br>
视解方程的结果，同一特征值的特征向量可能不同<br>其中部分实际上满足不同特征值的特征向量两两正<br>交化，而另一些不满足，严重增加计算量<br>应尽量避免Schmidt正交化<br>
构造可逆矩阵P=(<span class="equation-text" data-index="0" data-equation="\alpha_1,\alpha_2" contenteditable="false"><span></span><span></span></span>...)
写出P<span class="equation-text" data-index="0" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>AP=<span class="equation-text" data-index="1" data-equation="\Lambda" contenteditable="false"><span></span><span></span></span>
注意细节：特征值和<br>特征向量位置要匹配<br>
理论工具
向量内积
定义
<span class="equation-text" data-index="0" data-equation="[\vec x,\vec y]=\sum\limits_i x_iy_i" contenteditable="false"><span></span><span></span></span>
性质
[x,y]=[y,x]
[kx,y]=k[x,y]
[x+y,z]=[x,z]+[y,z]
x=0时[x,x]=0,x<span class="equation-text" data-index="0" data-equation="\neq" contenteditable="false"><span></span><span></span></span>0时,[x,x]>0<br>
施瓦茨不等式：[x,y]<span class="equation-text" data-index="0" data-equation="^2\leq" contenteditable="false"><span></span><span></span></span>[x,x][y,y]<br>
向量长度（范数）
定义<br>
<span class="equation-text" data-index="0" data-equation="||x||=\sqrt{[x,x]}" contenteditable="false"><span></span><span></span></span>
性质
非负性：||x||<span class="equation-text" data-index="0" data-equation="\geq" contenteditable="false"><span></span><span></span></span>0<br>
齐次性：||kx||=|k| ||x||<br>
向量的正交性
定义
[x,y]=0时，称x，y正交<br>
两两正交的向量构成正交向量组
n维单位向量<span class="equation-text" data-index="0" data-equation="e_i" contenteditable="false"><span></span><span></span></span>若是向量空间V的一个基，<br>且两两正交，称<span class="equation-text" data-index="1" data-equation="e_i" contenteditable="false"><span></span><span></span></span>是V的一个标准正交基<br>
性质
0与任何向量正交<br>
正交向量组中的向量线性无关
计算
n维单位向量<span class="equation-text" data-index="0" data-equation="a_i" contenteditable="false"><span></span><span></span></span>若是向量空间V的一个基，<br>要求V的一个标准正交基，即求一组<span class="equation-text" data-index="1" data-equation="e_i" contenteditable="false"><span></span><span></span></span>，<br>使<span class="equation-text" data-index="2" data-equation="e_i" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" data-index="3" data-equation="a_i" contenteditable="false"><span></span><span></span></span>等价，此过程称<span class="equation-text" data-index="4" data-equation="a_i" contenteditable="false"><span></span><span></span></span>的标准正交化<br>
施密特<br>正交化<br>
令<span class="equation-text" data-index="0" data-equation="b_k=a_k-\sum\limits_{i=1}^{k-1}\frac{[b_i,a_k]}{[b_i,b_i]}b_i" contenteditable="false"><span></span><span></span></span><br>再令<span class="equation-text" data-index="1" data-equation="e_k=\hat{b_k}" contenteditable="false"><span></span><span></span></span><br>
对于任何<span class="equation-text" data-index="0" data-equation="1\leq k\leq r" contenteditable="false"><span></span><span></span></span>，都满足<br>向量组<span class="equation-text" data-index="1" data-equation="b_1" contenteditable="false"><span></span><span></span></span>~<span class="equation-text" data-index="2" data-equation="b_k" contenteditable="false"><span></span><span></span></span>与<span class="equation-text" data-index="3" data-equation="a_1" contenteditable="false"><span></span><span></span></span>~<span class="equation-text" data-index="4" data-equation="a_k" contenteditable="false"><span></span><span></span></span>等价<br>
考察：求与给定向量组正交的向量
正交矩阵<br>
定义
若A满足A<span class="equation-text" data-index="0" data-equation="^T" contenteditable="false"><span></span><span></span></span>A=E，则A是正交矩阵
<font color="#F44336">充要条件：</font>A<span class="equation-text" data-index="0" data-equation="^T" contenteditable="false"><span></span><span></span></span>=A<span class="equation-text" data-index="1" data-equation="^{-1}" contenteditable="false"><span></span><span></span></span>
<font color="#F44336">充要条件：</font>A的列向量(或行向量)都是两两正交的单位向量
性质
若A为正交矩阵，则<span class="equation-text" data-index="0" data-equation="A^{-1}=A^T" contenteditable="false"><span></span><span></span></span>也是正交阵，且|A|<span class="equation-text" data-index="1" data-equation="^2" contenteditable="false"><span></span><span></span></span>=1<br>
若A，B是正交矩阵，则AB也是正交矩阵<br>
正交变换
定义
若P为正交阵，则线性变换y=Px称一个正交变换
性质
||y||=<span class="equation-text" data-index="0" data-equation="\sqrt{y^Ty}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\sqrt{x^TP^TPx}" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="2" data-equation="\sqrt{x^Tx}" contenteditable="false"><span></span><span></span></span>=||x||
正交变换下，长度不发生变化<br>
特征值与特征向量
定义
A是n阶矩阵。<span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>是n维<font color="#F44336">非零</font>列向量。满足A<span class="equation-text" data-index="1" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="2" data-equation="\lambda\alpha" contenteditable="false"><span></span><span></span></span><br>称<span class="equation-text" data-index="3" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>是矩阵A的特征值，<span class="equation-text" data-index="4" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>是A对应于<span class="equation-text" data-index="5" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>的特征向量<br>
移项，显然有(<span class="equation-text" data-index="0" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>E-A)<span class="equation-text" data-index="1" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>=0<br>
|<span class="equation-text" data-index="0" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>E-A|称为A的特征多项式，|<span class="equation-text" data-index="1" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>E-A|=0称A的特征方程
性质
if <span class="equation-text" data-index="0" data-equation="\alpha_i" contenteditable="false"><span></span><span></span></span>是<span class="equation-text" data-index="1" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>的特征向量，则任意不为零<br>的<span class="equation-text" data-index="2" data-equation="\sum\limits_ik_i\alpha_i" contenteditable="false"><span></span><span></span></span>也是<span class="equation-text" data-index="3" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>的特征向量<br>
if <span class="equation-text" data-index="0" data-equation="\alpha_1,\alpha_2" contenteditable="false"><span></span><span></span></span>分别是<span class="equation-text" data-index="1" data-equation="\lambda_1,\lambda_2" contenteditable="false"><span></span><span></span></span>的特征向量，<br>则<span class="equation-text" data-index="2" data-equation="\alpha_1+\alpha_2" contenteditable="false"><span></span><span></span></span>不是A的特征向量<br>
三角矩阵特征值是其对角线上的元素
<span class="equation-text" data-index="0" data-equation="\sum\lambda_i" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\sum a_{ii}" contenteditable="false"><span></span><span></span></span><br>
ie 矩阵的迹<br>
|A|=<span class="equation-text" data-index="0" data-equation="\prod\lambda_i" contenteditable="false"><span></span><span></span></span><br>
特征值不全为0<span class="equation-text" data-index="0" data-equation="\Leftrightarrow" contenteditable="false"><span></span><span></span></span>A可逆<br>
运算性质<br>设有A<span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\lambda\alpha" contenteditable="false"><span></span><span></span></span><br>
(A+kE)<span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="(\lambda+k)\alpha" contenteditable="false"><span></span><span></span></span>
A<span class="equation-text" data-index="0" data-equation="^n\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\lambda^n\alpha" contenteditable="false"><span></span><span></span></span>
A<span class="equation-text" data-index="0" data-equation="^n\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="A^{n-1}\lambda\alpha" contenteditable="false"><span></span><span></span></span><br>=<span class="equation-text" data-index="2" data-equation="\lambda A^{n-1}\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="3" data-equation="\lambda^n\alpha" contenteditable="false"><span></span><span></span></span><br>
A<span class="equation-text" data-index="0" data-equation="^{-1}\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\lambda^{-1}\alpha" contenteditable="false"><span></span><span></span></span>
A<span class="equation-text" data-index="0" data-equation="^*\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\frac{|A|}{\lambda}\alpha" contenteditable="false"><span></span><span></span></span>
计算
|<span class="equation-text" data-index="0" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>E-A|=0，可得n个特征值(<font color="#F44336">含重根</font>)。
根重数与线性无关特征向量的数量无关
一重根必定只有一个特征向量吗？
由定义A<span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\lambda\alpha" contenteditable="false"><span></span><span></span></span>
一般用于抽象矩阵
求解齐次方程组(<span class="equation-text" data-index="0" data-equation="\lambda" contenteditable="false"><span></span><span></span></span>E-A)x=0的<font color="#F44336">非0解</font>
(<span class="equation-text" data-index="0" data-equation="\lambda_i" contenteditable="false"><span></span><span></span></span>E-A)x=0基础解系即特征值<span class="equation-text" data-index="1" data-equation="\lambda_i" contenteditable="false"><span></span><span></span></span>的<br>特征向量的极大线性无关组<br>
方程组非0通解即特征值<span class="equation-text" data-index="0" data-equation="\lambda_i" contenteditable="false"><span></span><span></span></span>的全部特征向量
考察<br>
抽象矩阵求特征
考虑特征值、特征向量的内在联系
已知A的特征值为-1,0,4，<br>且A+B=2E，求B的特征值。<br>
对于3阶矩阵A，满足A<span class="equation-text" data-index="0" data-equation="^2" contenteditable="false"><span></span><span></span></span>+2A-3E=0<br>求证A的特征值只能是1或-3.<br>
不给出额外信息，无法确定具体的根重数以及根组成<br>
一般额外给出秩
具体矩阵求特征值<br>
(使用行列式、基础解系)
求<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}17&-2&-2\\-2&14&-4\\-2&-4&14\end{bmatrix}" contenteditable="false"><span></span><span></span></span>的特征。
<font color="#F44336">对于不含未知数的系数矩阵，求解过程中<br>行列式既然为0，可以直接消去一与其他行<br>成比例的行，再加减消元其余，以加快求解</font><br>
3阶矩阵就是三次方程，尽量先<br>得到一个解，避免求解3次方程<br>
可以构造只含1个非零项的行或<br>列以对行列式降阶
给出具体特征值/向量<br>求待定矩阵<br>
定义：A<span class="equation-text" data-index="0" data-equation="\alpha" contenteditable="false"><span></span><span></span></span>=<span class="equation-text" data-index="1" data-equation="\lambda\alpha" contenteditable="false"><span></span><span></span></span><br>
已知(1,1,-1)^T是矩阵<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}2&-1&-2\\5&a&3\\-1&b&-2\end{bmatrix}" contenteditable="false"><span></span><span></span></span>的特征向量，求a,b<br>
给出特征抽象组成，<br>求待定矩阵<br>
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