矩阵乘法说明.md

矩阵乘法：

$$
    \mathbf{A}_{m\times n} =
        \left[\begin{matrix}
            a_{11} & a_{12} & \cdots & a_{1n} \\
            a_{21} & a_{22} & \cdots & a_{2n} \\
            \vdots & \vdots & \ddots & \vdots \\
            a_{m1} & a_{m2} & \cdots & a_{mn} \\
        \end{matrix}\right], \quad
    \mathbf{B}_{n\times p} =
        \left[\begin{matrix}
            b_{11} & b_{12} & \cdots & b_{1p} \\
            b_{21} & b_{22} & \cdots & b_{2p} \\
            \vdots & \vdots & \ddots & \vdots \\
            b_{n1} & b_{n2} & \cdots & b_{np} \\
        \end{matrix}\right] \\[4em]
    \mathbf{C}_{m\times n} = \mathbf{A}_{m\times n} \mathbf{B}_{n\times p} =
        \left[\begin{matrix}
            \sum\limits_{k=1}^n a_{1k}b_{k1} & \sum\limits_{k=1}^n a_{1k}b_{k2} & \cdots & \sum\limits_{k=1}^n a_{1k}b_{kp} \\
            \sum\limits_{k=1}^n a_{2k}b_{k1} & \sum\limits_{k=1}^n a_{2k}b_{k2} & \cdots & \sum\limits_{k=1}^n a_{2k}b_{kp} \\
            \vdots & \vdots & \ddots & \vdots \\
            \sum\limits_{k=1}^n a_{mk}b_{k1} & \sum\limits_{k=1}^n a_{mk}b_{k2} & \cdots & \sum\limits_{k=1}^n a_{mk}b_{kp} \\
        \end{matrix}\right]
    \Longrightarrow
    C_{ij} = \sum_{k=1}^n a_{ik}b_{kj}
$$

```Python
from copy import deepcopy
from random import randint
import numpy as np

N = 3
P = 5
m = n = p = N

mat = [[randint(-5, 5) for n in range(N)] for m in range(N)]
print(f'M_{{{m}x{n}}} = ')
print(np.matrix(mat))

mul = deepcopy(mat)

for _ in range(P - 1):
    _mat = deepcopy(mat)
    # 提供 a_{ik}
    for i in range(m):
        # 提供 b_{kj}
        for j in range(p):
            mat[i][j] = sum(_mat[i][k] * mul[k][j] for k in range(n))

print(f'M ^ {P} = ')
print(np.matrix(mat))
print(f'(numpy) M ^ {P} = ')
print(np.matrix(mul) ** P)
```

```
M_{3x3} =
[[ 4  1 -5]
 [ 5  5  0]
 [-3 -3 -1]]

M ^ 5 =
[[ 24129  15921 -12300]
 [ 31530  20820 -16025]
 [-16995 -11226   8624]]

(numpy) M ^ 5 =
[[ 24129  15921 -12300]
 [ 31530  20820 -16025]
 [-16995 -11226   8624]]
```