【数理知识】矩阵普通乘积，哈达玛积，克罗内克积，点乘，点积，叉乘，matlab代码实现-Toy模板网

这篇具有很好参考价值的文章主要介绍了【数理知识】矩阵普通乘积，哈达玛积，克罗内克积，点乘，点积，叉乘，matlab代码实现。希望对大家有所帮助。如果存在错误或未考虑完全的地方，请大家不吝赐教，您也可以点击"举报违法"按钮提交疑问。

序号	内容
1	【数理知识】向量数乘，内积，外积，matlab代码实现
2	【数理知识】矩阵普通乘积，哈达玛积，克罗内克积，点乘，点积，叉乘，matlab代码实现

首先介绍矩阵

1. 矩阵基本形式

在数学中，矩阵是一个按照长方阵列排列的复数或实数集合。由 $\times n$ 个数 $a_{ij}$ 排成的 $m$ 行 $n$ 列的数表称为 $m$ 行 $n$ 列的矩阵，简称 $\times n$ 矩阵。记作：

$\left[\begin{matrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ a_{m1} & a_{m2} & \cdots & a_{mn} \\ \end{matrix}\right]$

2. 矩阵基本运算 - 普通乘积，matmul product

矩阵相乘最重要的方法是一般矩阵乘积。它只有在第一个矩阵的列数（column）和第二个矩阵的行数（row）相同时才有意义。即线性代数学学的，左行乘以右列。

左边矩阵的列数必须与右边矩阵的行数相等
不满足交换律，即 $\ne BA (BA 存在)$

$\left[\begin{matrix} \red{a_{11}} & \red{a_{12}} & \red{a_{13}} \\ a_{21} & a_{22} & a_{23} \\ \end{matrix}\right]_{\green{2 \times 3}}, \quad B = \left[\begin{matrix} \blue{b_{11}} & b_{12} & b_{13} & b_{14} \\ \blue{b_{21}} & b_{22} & b_{23} & b_{24} \\ \blue{b_{31}} & b_{32} & b_{33} & b_{34} \\ \end{matrix}\right]_{\green{3 \times 4}}$

$\begin{aligned} C &= A_{\green{2 \times 3}} B_{\green{3 \times 4}} = C_{\green{2 \times 4}} \\ &=\left[\begin{matrix} \red{a_{11}} \blue{b_{11}} + \red{a_{12}} \blue{b_{21}} + \red{a_{13}} \blue{b_{31}} & \red{a_{11}} b_{12} + \red{a_{12}} b_{22} + \red{a_{13}} b_{32} & \red{a_{11}} b_{13} + \red{a_{12}} b_{23} + \red{a_{13}} b_{33} & \red{a_{11}} b_{14} + \red{a_{12}} b_{24} + \red{a_{13}} b_{34} \\ a_{21} \blue{b_{11}} + a_{22} \blue{b_{21}} + a_{23} \blue{b_{31}} & a_{21} b_{12} + a_{22} b_{22} + a_{23} b_{32} & a_{21} b_{13} + a_{22} b_{23} + a_{23} b_{33} & a_{21} b_{14} + a_{22} b_{24} + a_{23} b_{34} \\ \end{matrix}\right]_{\green{2 \times 4}} \end{aligned}$

A = [1  2  3; 
	 4  5  6];
B = [1  2  3  4; 
	 5  6  7  8; 
	 9 10 11 12];

Matlab语法：C = A * B

>> C = A * B
ans =
    38    44    50    56
    83    98   113   128

>> A.*B
对于此运算，数组的大小不兼容。

>> dot(A, B)
错误使用 dot
A 和 B 的大小必须相同。
 
>> cross(A, B)
错误使用 cross
A 和 B 的大小必须相同。

3. 矩阵基本运算 - 哈达玛积 Hadamard product

当矩阵 $A$ 和矩阵 $B$ 的维度相同时，矩阵点乘即为哈达玛积（Hadamard Product/Point-wise Product/Element-wise Product/Element-wise Multiplication）

哈达玛积其元素定义为两个矩阵对应元素的乘积
矩阵各个对应元素相乘, 这个时候要求两个矩阵必须同样大小

$\left[\begin{matrix} \red{a_{11}} & \red{a_{12}} & \red{a_{13}} \\ a_{21} & a_{22} & a_{23} \\ a_{31} & a_{32} & a_{33} \\ \end{matrix}\right]_{\green{3 \times 3}}, \quad B = \left[\begin{matrix} \blue{b_{11}} & b_{12} & b_{13} \\ \blue{b_{21}} & b_{22} & b_{23} \\ \blue{b_{31}} & b_{32} & b_{33} \\ \end{matrix}\right]_{\green{3 \times 3}}$

$\begin{aligned} C &= A_{\green{3 \times 3}} B_{\green{3 \times 3}} = C_{\green{3 \times 3}} \\ &=\left[\begin{matrix} \red{a_{11}} \blue{b_{11}} & \red{a_{12}} b_{12} & \red{a_{13}} b_{13} \\ a_{21} \blue{b_{21}} & a_{22} b_{22} & a_{23} b_{23} \\ a_{31} \blue{b_{31}} & a_{32} b_{32} & a_{33} b_{33} \\ \end{matrix}\right]_{\green{3 \times 3}} \end{aligned}$

A = [1  2  3; 
	 4  5  6;
     7  8  9];
B = [1  2  3; 
	 4  5  6;
     7  8  9];

Matlab语法：C = A .* B

>> C = A .* B
C =
     1     4     9
    16    25    36
    49    64    81

>> C = A * B
C =
    30    36    42
    66    81    96
   102   126   150

>> C = dot(A, B)
C =
    66    93   126
    
>> C = cross(A, B)
C =
     0     0     0
     0     0     0
     0     0     0

4. 矩阵基本运算 - 克罗内克积，Kronecker product

$\left[\begin{matrix} \red{a_{11}} & \red{a_{12}} & \red{a_{13}} \\ a_{21} & a_{22} & a_{23} \\ \end{matrix}\right]_{\green{2 \times 3}}, \quad B = \left[\begin{matrix} \blue{b_{11}} & b_{12} \\ \blue{b_{21}} & b_{22} \\ \blue{b_{31}} & b_{32} \\ \end{matrix}\right]_{\green{3 \times 2}}$

$\begin{aligned} C &= A_{\green{2 \times 3}} B_{\green{3 \times 2}} = C_{\green{6 \times 6}} \\ &=\left[\begin{matrix} \red{a_{11}} \blue{b_{11}} & \red{a_{11}} b_{12} & \red{a_{12}} \blue{b_{11}} & \red{a_{12}} b_{12} & \red{a_{13}} \blue{b_{11}} & \red{a_{13}} b_{12} \\ \red{a_{11}} \blue{b_{21}} & \red{a_{11}} b_{22} & \red{a_{12}} \blue{b_{21}} & \red{a_{12}} b_{22} & \red{a_{13}} \blue{b_{21}} & \red{a_{13}} b_{22} \\ \red{a_{11}} \blue{b_{31}} & \red{a_{11}} b_{32} & \red{a_{12}} \blue{b_{31}} & \red{a_{12}} b_{32} & \red{a_{13}} \blue{b_{31}} & \red{a_{13}} b_{32} \\ a_{21} \blue{b_{11}} & a_{21} b_{12} & a_{22} \blue{b_{11}} & a_{22} b_{12} & a_{23} \blue{b_{11}} & a_{23} b_{12} \\ a_{21} \blue{b_{21}} & a_{21} b_{22} & a_{22} \blue{b_{21}} & a_{22} b_{22} & a_{23} \blue{b_{21}} & a_{23} b_{22} \\ a_{21} \blue{b_{31}} & a_{21} b_{32} & a_{22} \blue{b_{31}} & a_{22} b_{32} & a_{23} \blue{b_{31}} & a_{23} b_{32} \\ \end{matrix}\right]_{\green{6 \times 6}} \end{aligned}$

A = [1  2  3; 
	 4  5  6];
B = [1  2; 
	 3  4;
     5  6];

Matlab语法：C = kron(A, B)

>> C = kron(A, B)
C =
     1     2     2     4     3     6
     3     4     6     8     9    12
     5     6    10    12    15    18
     4     8     5    10     6    12
    12    16    15    20    18    24
    20    24    25    30    30    36
    
>> C = A * B
C =
    22    28
    49    64

>> C = dot(A, B)
错误使用 dot
A 和 B 的大小必须相同。

>> C = cross(A, B)
错误使用 cross
A 和 B 的大小必须相同。

5. Matlab矩阵运算 - 普通乘积 *

这个比较简单，不再过多解释。

6. Matlab矩阵运算 - 点乘 .*

$\left[\begin{matrix} \red{a_{11}} & \red{a_{12}} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \\ a_{41} & a_{42} \\ \end{matrix}\right]_{\green{4 \times 2}}, \quad B = \left[\begin{matrix} \blue{b_{11}} \\ \blue{b_{21}} \\ \blue{b_{31}} \\ \blue{b_{41}} \\ \end{matrix}\right]_{\green{4 \times 1}}$

$\begin{aligned} C &= A_{\green{4 \times 2}} \odot B_{\green{4 \times 1}} = C_{\green{4 \times 2}} \\ &=\left[\begin{matrix} \red{a_{11}} \blue{b_{11}} & \red{a_{12}} \blue{b_{11}} \\ a_{21} \blue{b_{21}} & a_{22} \blue{b_{21}} \\ a_{31} \blue{b_{31}} & a_{32} \blue{b_{31}} \\ a_{41} \blue{b_{41}} & a_{42} \blue{b_{41}} \\ \end{matrix}\right]_{\green{4 \times 2}} \end{aligned}$

A = [1  2; 
	 3  4;
     5  6;
     7  8];
B = [1; 
	 2;
     3;
     4];

>> C = A .* B
C =
     1     2
     6     8
    15    18
    28    32

$\left[\begin{matrix} \red{a_{11}} \\ a_{21} \\ \end{matrix}\right]_{\green{2 \times 1}}, \quad B = \left[\begin{matrix} \blue{b_{11}} & b_{12} & b_{13} \\ \end{matrix}\right]_{\green{1 \times 3}}$

$\begin{aligned} C &= A_{\green{2 \times 1}} \odot B_{\green{1 \times 3}} = C_{\green{2 \times 3}} \\ &=\left[\begin{matrix} \red{a_{11}} \blue{b_{11}} & \red{a_{11}} \blue{b_{12}} & \red{a_{11}} \blue{b_{13}} \\ a_{21} \blue{b_{11}} & a_{21} b_{12} & a_{21} b_{13} \\ \end{matrix}\right]_{\green{2 \times 3}} \end{aligned}$

A = [1;
	 2];
B = [1  2  3];

>> C = A .* B
C =
     1     2     3
     2     4     6

$\left[\begin{matrix} \red{a_{11}} \\ a_{21} \\ \end{matrix}\right]_{\green{2 \times 1}}, \quad B = \left[\begin{matrix} \blue{b_{11}} & b_{12} & b_{13} \\ \blue{b_{21}} & b_{22} & b_{23} \\ \end{matrix}\right]_{\green{2 \times 3}}$

$\begin{aligned} C &= A_{\green{2 \times 1}} \odot B_{\green{2 \times 3}} = C_{\green{2 \times 3}} \\ &=\left[\begin{matrix} \red{a_{11}} \blue{b_{11}} & \red{a_{11}} \blue{b_{12}} & \red{a_{11}} \blue{b_{13}} \\ a_{21} \blue{b_{12}} & a_{21} b_{22} & a_{21} b_{23} \\ \end{matrix}\right]_{\green{2 \times 3}} \end{aligned}$

A = [1;
	 2];
B = [1  2  3;
	 4  5  6];

>> C = A .* B
C =
     1     2     3
     8    10    12

7. Matlab矩阵运算 - 点积 dot()

C = dot(A,B) returns the scalar dot product of A and B.

If A and B are vectors, then they must have the same length.
If A and B are matrices or multidimensional arrays, then they must have the same size. In this case, the dot function treats A and B as collections of vectors. The function calculates the dot product of corresponding vectors along the first array dimension whose size does not equal 1.

C = dot(A,B,dim) evaluates the dot product of A and B along dimension, dim. The dim input is a positive integer scalar.

8. Matlab矩阵运算 - 叉乘 cross()

C = cross(A,B) returns the cross product of A and B.

If A and B are vectors, then they must have a length of 3.
If A and B are matrices or multidimensional arrays, then they must have the same size. In this case, the cross function treats A and B as collections of three-element vectors. The function calculates the cross product of corresponding vectors along the first array dimension whose size equals 3.

C = cross(A,B,dim) evaluates the cross product of arrays A and B along dimension, dim. A and B must have the same size, and both size(A,dim) and size(B,dim) must be 3. The dim input is a positive integer scalar.文章来源地址https://www.toymoban.com/news/detail-443452.html