
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

文章目录

1. 矩阵基本形式
2. 矩阵基本运算 - 普通乘积，matmul product
3. 矩阵基本运算 - 哈达玛积 Hadamard product
4. 矩阵基本运算 - 克罗内克积，Kronecker product
5. Matlab矩阵运算 - 普通乘积 *
6. Matlab矩阵运算 - 点乘 .*
7. Matlab矩阵运算 - 点积 dot()
8. Matlab矩阵运算 - 叉乘 cross()
Ref

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 m×nm \times nm×n个数 aija_{ij}aij 排成的 mmm 行 nnn 列的数表称为 mmm 行 nnn 列的矩阵，简称 m×nm \times nm×n 矩阵。记作：

A=[a11a12⋯a1na21a22⋯a2n⋮⋮⋱⋮am1am2⋯amn]A = \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]A=a11a21⋮am1a12a22⋮am2⋯⋯⋱⋯a1na2n⋮amn

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 AB≠BA(BA存在)AB \ne BA (BA 存在)AB=BA(BA存在)

A=[a11a12a13a21a22a23]2×3,B=[b11b12b13b14b21b22b23b24b31b32b33b34]3×4A = \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}}A=[a11a21a12a22a13a23]2×3,B=b11b21b31b12b22b32b13b23b33b14b24b343×4

C=A2×3B3×4=C2×4=[a11b11+a12b21+a13b31a11b12+a12b22+a13b32a11b13+a12b23+a13b33a11b14+a12b24+a13b34a21b11+a22b21+a23b31a21b12+a22b22+a23b32a21b13+a22b23+a23b33a21b14+a22b24+a23b34]2×4\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}C=A2×3B3×4=C2×4=[a11b11+a12b21+a13b31a21b11+a22b21+a23b31a11b12+a12b22+a13b32a21b12+a22b22+a23b32a11b13+a12b23+a13b33a21b13+a22b23+a23b33a11b14+a12b24+a13b34a21b14+a22b24+a23b34]2×4

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    5683    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 AAA 和矩阵 BBB 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

A=[a11a12a13a21a22a23a31a32a33]3×3,B=[b11b12b13b21b22b23b31b32b33]3×3A = \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}}A=a11a21a31a12a22a32a13a23a333×3,B=b11b21b31b12b22b32b13b23b333×3

C=A3×3B3×3=C3×3=[a11b11a12b12a13b13a21b21a22b22a23b23a31b31a32b32a33b33]3×3\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}C=A3×3B3×3=C3×3=a11b11a21b21a31b31a12b12a22b22a32b32a13b13a23b23a33b333×3

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     916    25    3649    64    81>> C = A * B
C =30    36    4266    81    96102   126   150>> C = dot(A, B)
C =66    93   126>> C = cross(A, B)
C =0     0     00     0     00     0     0

4. 矩阵基本运算 - 克罗内克积，Kronecker product

A=[a11a12a13a21a22a23]2×3,B=[b11b12b21b22b31b32]3×2A = \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}}A=[a11a21a12a22a13a23]2×3,B=b11b21b31b12b22b323×2

C=A2×3B3×2=C6×6=[a11b11a11b12a12b11a12b12a13b11a13b12a11b21a11b22a12b21a12b22a13b21a13b22a11b31a11b32a12b31a12b32a13b31a13b32a21b11a21b12a22b11a22b12a23b11a23b12a21b21a21b22a22b21a22b22a23b21a23b22a21b31a21b32a22b31a22b32a23b31a23b32]6×6\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}C=A2×3B3×2=C6×6=a11b11a11b21a11b31a21b11a21b21a21b31a11b12a11b22a11b32a21b12a21b22a21b32a12b11a12b21a12b31a22b11a22b21a22b31a12b12a12b22a12b32a22b12a22b22a22b32a13b11a13b21a13b31a23b11a23b21a23b31a13b12a13b22a13b32a23b12a23b22a23b326×6

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     63     4     6     8     9    125     6    10    12    15    184     8     5    10     6    1212    16    15    20    18    2420    24    25    30    30    36>> C = A * B
C =22    2849    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 .*

A=[a11a12a21a22a31a32a41a42]4×2,B=[b11b21b31b41]4×1A = \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}}A=a11a21a31a41a12a22a32a424×2,B=b11b21b31b414×1

C=A4×2⊙B4×1=C4×2=[a11b11a12b11a21b21a22b21a31b31a32b31a41b41a42b41]4×2\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}C=A4×2⊙B4×1=C4×2=a11b11a21b21a31b31a41b41a12b11a22b21a32b31a42b414×2

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

>> C = A .* B
C =1     26     815    1828    32

A=[a11a21]2×1,B=[b11b12b13]1×3A = \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}}A=[a11a21]2×1,B=[b11b12b13]1×3

C=A2×1⊙B1×3=C2×3=[a11b11a11b12a11b13a21b11a21b12a21b13]2×3\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}C=A2×1⊙B1×3=C2×3=[a11b11a21b11a11b12a21b12a11b13a21b13]2×3

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

>> C = A .* B
C =1     2     32     4     6

A=[a11a21]2×1,B=[b11b12b13b21b22b23]2×3A = \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}}A=[a11a21]2×1,B=[b11b21b12b22b13b23]2×3

C=A2×1⊙B2×3=C2×3=[a11b11a11b12a11b13a21b12a21b22a21b23]2×3\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}C=A2×1⊙B2×3=C2×3=[a11b11a21b12a11b12a21b22a11b13a21b23]2×3

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

>> C = A .* B
C =1     2     38    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.

Ref

矩阵和向量的点乘与叉乘
几种矩阵乘法总结
矩阵乘法，还可以这样算？
向量和矩阵的点乘和叉乘

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