线性代数（矩阵和向量）

川长思鸟来 2023-02-17 01:16 6阅读 0赞

### 目录 ###

*  矩阵和向量
 *  加法和标量乘法
 *  矩阵向量乘法
 *  矩阵乘法
 *  矩阵乘法特征
 *  逆和转置

# 矩阵和向量 #

**矩阵：**  
下图为一个R(4x2)和一个R(2x3)的矩阵  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 1]  
矩阵优点：快速整理，索引和访问大量数据。

**向量：**  
下图为一个R(4)的向量  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 2]

# 加法和标量乘法 #

**加法：**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 3]  
矩阵加法是逐个元素相加，只有维度相同的才能相加。  
**标量乘法：**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 4]  
标量乘法也是与矩阵的逐个元素进行运算。  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 5]

# 矩阵向量乘法 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 6]  
矩阵向量乘法必须前一个矩阵的列数等于第二个向量的行数，运算结果：R(mxn) \* R(nx1) = R(mx1)。  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 7]  
**技巧：**  
以房价预测为例：  
假设有四个房子待预测，可以转换成如下方式经行计算  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 8]  
优点：代码更简洁，计算效率更高。

# 矩阵乘法 #

![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 9]  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 10]  
矩阵乘法必须前一个矩阵的列数等于第二个矩阵的行数，运算结果：R(mxn) \* R(nxo) = R(mxo)。  
**技巧：**  
以预测房价为例：  
假设有四间房子，三个假设，转化成两个矩阵进行计算  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 11]  
利用编程语言的线性代数库进行高效的矩阵乘法运算。

# 矩阵乘法特征 #

**矩阵乘法不服从交换律**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 12]  
**矩阵乘法服从结合律**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 13]  
**单位矩阵：**

用 I 表示单位矩阵  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 14]

# 逆和转置 #

**求逆矩阵：**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 15]  
只有方阵才有逆矩阵（如果所以元素都为0也没有逆矩阵）。  
无逆矩阵的称为奇异矩阵（退化矩阵）。  
在Octave中，假设方阵A，求它的逆矩阵，输入语句 pinv(A) 即可求出逆矩阵。

**转置矩阵：**  
![在这里插入图片描述][watermark_type_ZmFuZ3poZW5naGVpdGk_shadow_10_text_aHR0cHM6Ly9ibG9nLmNzZG4ubmV0L3dlaXhpbl80NTczNTI0Mg_size_16_color_FFFFFF_t_70 16]

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