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LA-2-矩阵与运算

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LA-2-矩阵与运算

2021-08-07 16:57:38   18  举报

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线性代数第二章、第三章(部分)矩阵及其运算知识点梳理

线性代数

考研

矩阵

考研数学

大学教育

模版推荐

作者其他创作

大纲/内容

分块矩阵

运算法则

四则同矩阵

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}A&B\\C&D\end{bmatrix}^T=\begin{bmatrix}A^T&C^T\\B^T&D^T\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}A&0\\0&B\end{bmatrix}^n=\begin{bmatrix}A^n&0\\0&B^n\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

应用

<span class="equation-text" data-index="0" data-equation="\left[\begin{array}{cc:c}&&\\&&\\\hdashline&&\end{array}\right]" contenteditable="false"><span></span><span></span></span><br>分4块的应用<br>

矩阵相乘

矩阵的幂

逆矩阵<br>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}A&B\\C&D\end{bmatrix}^{-1}=\frac 1{AD-BC}\begin{bmatrix}D&-B\\-C&A\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}A&0\\0&B\end{bmatrix}^{-1}=\begin{bmatrix}A^{-1}&0\\0&B^{-1}\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}0&A\\B&0\end{bmatrix}^{-1}=\begin{bmatrix}0&B^{-1}\\A^{-1}&0\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\left[\begin{array}{ccc}&&\\\hdashline&&\\\hdashline&&\end{array}\right]" contenteditable="false"><span></span><span></span></span><br>行分块的应用<br>

向量，秩

<span class="equation-text" data-index="0" data-equation="\left[\begin{array}{c:c:c}&&\\&&\\&&\end{array}\right]" contenteditable="false"><span></span><span></span></span><br>列分块的应用<br>

向量，秩

解方程组

对于AB=C<br>

如A可逆则<span class="equation-text" data-index="0" data-equation="A^{-1}C=B" contenteditable="false"><span></span><span></span></span><br>

B的行向量可由C的行向量线性表出<br><span class="equation-text" data-index="0" data-equation="A^{-1}\begin{bmatrix}\gamma_1&\gamma_2&\gamma_3\end{bmatrix}^T=\begin{bmatrix}\beta_1&\beta_2&\beta_3\end{bmatrix}^T" contenteditable="false"><span></span><span></span></span>

如B可逆则<span class="equation-text" data-index="0" data-equation="CB^{-1}=A" contenteditable="false"><span></span><span></span></span><br>

A的列向量可由C的列向量线性表出<br><span class="equation-text" data-index="0" data-equation="\begin{bmatrix}\gamma_1&\gamma_2&\gamma_3\end{bmatrix}B^{-1}=\begin{bmatrix}\alpha_1&\alpha_2&\alpha_3\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

对于AB=0<br>

B的行向量组成的列向量是方程的解<br>

例<br>

设A=<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}1&2&-2\\4&t&3\\3&-1&1\end{bmatrix}" contenteditable="false"><span></span><span></span></span>,B为3阶非0矩阵，且AB=0，求t

方阵的行列式<br>

<span class="equation-text" data-index="0" data-equation="|A^T|=|A|" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="|kA|=k^n|A|" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="|AB|=|A||B|" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="|A^2|=|A|^2" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="|A^*|=|A|^{n-1}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="|A^{-1}|=|A|^{-1}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="|A^{-1}|=\frac 1{|A|}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{vmatrix}A&O\\C&B\end{vmatrix}=\begin{vmatrix}A&D\\O&B\end{vmatrix}=|A||B|" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{vmatrix}O&A\\B&C\end{vmatrix}=\begin{vmatrix}D&A\\B&O\end{vmatrix}=(-1)^{mn}|A||B|" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="if A \sim B,|A|=|B|" contenteditable="false"><span></span><span></span></span>

一般情况下<br>

<span class="equation-text" data-index="0" data-equation="|A\pm B|\neq|A|\pm |B|" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="|kA|\neq k|A|" contenteditable="false"><span></span><span></span></span>

矩阵

基本概念及运算法则

特征

秩(Rank)<br>

矩阵中任取k行k列，形成的k<span class="equation-text" data-index="0" data-equation="\times" contenteditable="false"><span></span><span></span></span>k大小的区域构成的行列式称为矩阵的k阶子式<br>

非0子式的最高阶数称为矩阵的秩

初等变换不改变矩阵的秩

行阶梯矩阵的秩等于其非0行数

结论<br>

0矩阵的秩为0<br>

可逆矩阵的秩等于其阶数

不可逆矩阵的秩小于其阶数

<span class="equation-text" data-index="0" data-equation="r(A_{m\times n})\in[0,min\{m,n\}]" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="r(A^T)=r(A)" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="若A\sim B,r(A)=r(B)" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="若PQ可逆，r(PAQ)=r(A)" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="r(A,B)\in[max\{r(A),r(B)\},r(A)+r(B)]" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="r(A+B)\leq r(A)+r(B)" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="r(AB)\leq min\{r(A),r(B)\}" contenteditable="false"><span></span><span></span></span><br>

<span class="equation-text" data-index="0" data-equation="若A_{m\times n}B_{n\times l}=0,则r(A)+r(B)\leq n" contenteditable="false"><span></span><span></span></span><br>

伴随矩阵

最重要的性质：<span class="equation-text" data-index="0" data-equation="A^*A=AA^*=|A|E" contenteditable="false"><span></span><span></span></span>，即<span class="equation-text" data-index="1" data-equation="A^{-1}=\frac{A^*}{|A|}" contenteditable="false"><span></span><span></span></span><br>

<span class="equation-text" data-index="0" data-equation="\color{red}注意顺序！根据具体情况选择A和A^*的先后" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\vec X=C_1\begin{bmatrix}1\\0\\1\end{bmatrix}+\ C_2\begin{bmatrix}2\\4\\-4\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="(A^*)^{-1}=(A^{-1})^*=\frac 1{|A|}A" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="r(A^*)=\left\{\begin{aligned}&n,&若&r(A)=n\\&1,&若&r(A)=n-1\\&0,&若&r(A)<n-1\end{aligned}\right." contenteditable="false"><span></span><span></span></span>

逆矩阵

矩阵A可逆<span class="equation-text" data-index="0" data-equation="\Leftrightarrow det(A)\neq0" contenteditable="false"><span></span><span></span></span><br>

可逆矩阵

可逆矩阵的秩等于其阶数

又称为满秩矩阵

求法<span class="equation-text" data-index="0" data-equation="(A,E)→...→(E,A^{-1})" contenteditable="false"><span></span><span></span></span>

行阶梯矩阵

满足

矩阵若有0行，则0行在矩阵底部

每个非零行的主元(首个非零元)<br>下方只有0元素或无<br>

行最简矩阵

满足

是行阶梯矩阵

每个非零行的主元必须是1<br>所在列其他元素均为0<br>

用于解方程组：方程组的增广矩阵<span class="equation-text" data-index="0" data-equation="\bar A" contenteditable="false"><span></span><span></span></span>化为行最简<br>

考察：<br>

将矩阵化为行阶梯、行最简矩阵

相似矩阵<br><span class="equation-text" data-index="0" data-equation="A\sim B" contenteditable="false"><span></span><span></span></span><br>

若存在可逆矩阵P，使得<span class="equation-text" data-index="0" data-equation="P^{-1}AP=B" contenteditable="false"><span></span><span></span></span>，称A与B相似，记为<span class="equation-text" data-index="1" data-equation="A\sim B" contenteditable="false"><span></span><span></span></span><br>

等价矩阵<br><span class="equation-text" data-index="0" data-equation="A\cong B" contenteditable="false"><span></span><span></span></span>

A经过有限次初等变换得到B，称A与B等价，记为<span class="equation-text" data-index="0" data-equation="A\cong B" contenteditable="false"><span></span><span></span></span><br>

等价矩阵形状相同<br>

<span class="equation-text" data-index="0" data-equation="A\cong B \Leftrightarrow r(A)=r(B)" contenteditable="false"><span></span><span></span></span><br>

等价标准型<br>

设<span class="equation-text" data-index="0" data-equation="A_{m\times n},\exists" contenteditable="false"><span></span><span></span></span>可逆矩阵<span class="equation-text" data-index="1" data-equation="P_{m\times m},Q_{n\times n}" contenteditable="false"><span></span><span></span></span>

使<span class="equation-text" data-index="0" data-equation="PAQ=\begin{bmatrix}E_r&0\\0&0\end{bmatrix}" contenteditable="false"><span></span><span></span></span><br>

称PAQ得到的矩阵为A的等价标准型

矩阵的初等(行/列)变换

初等行/列变换

初等变换的深刻意义在于：初等变换不改变矩阵的秩

倍乘(非0元)

互换

倍加

初等矩阵<br><font color="#F44336">单位矩阵</font>经过一次初<br>等变换得到的矩阵<br>

例<br>

<span class="equation-text" data-index="0" data-equation="\begin{pmatrix}k&0&0\\0&1&0\\0&0&1\end{pmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{pmatrix}0&1&0\\1&0&0\\0&0&1\end{pmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{pmatrix}1&0&0\\k&1&0\\0&0&1\end{pmatrix}" contenteditable="false"><span></span><span></span></span>

初等矩阵通常可按“经历一次行变换”<br>“经历一次列变换”两种方式构成<br>

初等矩阵P左乘矩阵A，即AP，相当<br>于A做一次与P相同的初等行变换<br>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}1&0\\0&-1\end{bmatrix}\begin{bmatrix}1&2\\3&4\end{bmatrix}=\begin{bmatrix}1&2\\-3&-4\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

初等矩阵P右乘于矩阵A，即PA，相<br>当于A做一次与P相同的初等列变换<br>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}1&2\\3&4\end{bmatrix}\begin{bmatrix}1&0\\0&-1\end{bmatrix}=\begin{bmatrix}1&-2\\3&-4\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

若<span class="equation-text" data-index="0" data-equation="P_t\cdots P_2P_1A=B," contenteditable="false"><span></span><span></span></span>记<span class="equation-text" data-index="1" data-equation="P=P_t\cdots P_2P_1" contenteditable="false"><span></span><span></span></span><br>其中P为初等矩阵，则必有<span class="equation-text" data-index="2" data-equation="P_t\cdots P_2P_1E=P" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="(A,E)→...→(B,P)" contenteditable="false"><span></span><span></span></span>

推论:<span class="equation-text" data-index="0" data-equation="(A,E)→...→(E,A^{-1})" contenteditable="false"><span></span><span></span></span>

列变换类似，不过分块矩阵要竖着构造<br>

初等矩阵的逆矩阵为其逆变换对应的矩阵<br>初等矩阵可逆，且其逆与其同类<br>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}k&0&0\\0&1&0\\0&0&1\end{bmatrix}^{-1}=\begin{bmatrix}\frac 1 k&0&0\\0&1&0\\0&0&1\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}^{-1}=\begin{bmatrix}0&1&0\\1&0&0\\0&0&1\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

<span class="equation-text" data-index="0" data-equation="\begin{bmatrix}1&0&0\\k&1&0\\0&0&1\end{bmatrix}^{-1}=\begin{bmatrix}1&0&0\\-k&1&0\\0&0&1\end{bmatrix}" contenteditable="false"><span></span><span></span></span>

考察方式

给出原始矩阵和结果矩阵，求中间变换对应的初等矩阵

给出初等矩阵原始矩阵，求结果矩阵

给出初等矩阵和结果矩阵，求原始矩阵

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