概述
向量是对数的拓展,一个向量表示一组数;而矩阵则可以视为对向量的拓展,一个矩阵表示一组向量。
看待一个矩阵有两个视角,行向量视角和列向量视角。
当行数和列数相等时候,称为方阵,方阵有很多特殊的性质。有很多特殊的性质的矩阵,是方阵。
实现矩阵类
from vector import Vector
class Matrix:
def __init__(self, list2d):
self._value = list2d.copy()
def __repr__(self):
return "Matrix({})".format(self._value)
# 此处简单设置交互式与print模式打印的内容相同
__str__ = __repr__
def shape(self):
return len(self._value), len(self._value[0])
def row_num(self):
"""返回矩阵行数"""
return self.shape()[0]
def col_num(self):
"""返回矩阵列数"""
return self.shape()[1]
def size(self):
"""矩阵元素个数"""
r, c = self.shape()
return r * c
def __getitem__(self, pos):
"""返回指定未知数的元素,pos的形式是元组"""
assert pos[0] < self.row_num() and pos[1] < self.col_num
r, c = pos
return self._value[r][c]
def row_vector(self, idx):
"""返回第idx个行向量"""
return Vector(self._value[idx])
def col_vector(self, idx):
"""返回第idx个列向量"""
return Vector([row[idx] for row in self._value)
矩阵基本运算和性质
矩阵加法
两个同形矩阵加法,
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A=\begin{pmatrix}a_{11}&a_{12}&...&a_{1c}\\a_{21}&a_{22}&...&a_{2c}\\...&...&...&...\\a_{r1}&a_{r2}&...&a_{rc}\end{pmatrix}B=\begin{pmatrix}b_{11}&b_{12}&...&b_{1c}\\b_{21}&b_{22}&...&b_{2c}\\...&...&...&...\\b_{r1}&b_{r2}&...&b_{rc}\end{pmatrix}
A=⎝⎜⎜⎛a11a21...ar1a12a22...ar2............a1ca2c...arc⎠⎟⎟⎞B=⎝⎜⎜⎛b11b21...br1b12b22...br2............b1cb2c...brc⎠⎟⎟⎞
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A+B=\begin{pmatrix}a_{11}+b_{11}&a_{12}+b_{12}&...&a_{1c}+b_{1c}\\a_{21}+b_{21}&a_{22}+b_{22}&...&a_{2c}+b_{2c}\\...&...&...&...\\a_{r1}+b_{r1}&a_{r2}+b_{r2}&...&a_{rc}+b_{rc}\end{pmatrix}
A+B=⎝⎜⎜⎛a11+b11a21+b21...ar1+br1a12+b12a22+b22...ar2+br2............a1c+b1ca2c+b2c...arc+brc⎠⎟⎟⎞
矩阵数乘
一个实数与一个矩阵的乘法运算,
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A=\begin{pmatrix}a_{11}&a_{12}&...&a_{1c}\\a_{21}&a_{22}&...&a_{2c}\\...&...&...&...\\a_{r1}&a_{r2}&...&a_{rc}\end{pmatrix}
A=⎝⎜⎜⎛a11a21...ar1a12a22...ar2............a1ca2c...arc⎠⎟⎟⎞
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k\cdot A=\begin{pmatrix}k\cdot a_{11}&k\cdot a_{12}&...&k\cdot a_{1c}\\k\cdot a_{21}&k\cdot a_{22}&...&k\cdot a_{2c}\\...&...&...&...\\k\cdot a_{r1}&k\cdot a_{r2}&...&k\cdot a_{rc}\end{pmatrix}
k⋅A=⎝⎜⎜⎛k⋅a11k⋅a21...k⋅ar1k⋅a12k⋅a22...k⋅ar2............k⋅a1ck⋅a2c...k⋅arc⎠⎟⎟⎞
矩阵运算性质
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交换律
A + B = B + A A + B = B + A A+B=B+A -
结合律
( A + B ) + C = A + ( B + C ) (A + B) + C = A + (B + C) (A+B)+C=A+(B+C)
( c ⋅ k ) ⋅ A = c ⋅ ( k ⋅ A ) (c\cdot k) \cdot A = c\cdot(k\cdot A) (c⋅k)⋅A=c⋅(k⋅A) 。其中c和k是实数
k ⋅ ( A + B ) = k ⋅ A + k ⋅ B k\cdot(A + B) = k\cdot A + k\cdot B k⋅(A+B)=k⋅A+k⋅B
( c + k ) ⋅ A = c ⋅ A + k ⋅ A (c + k) \cdot A = c\cdot A + k\cdot A (c+k)⋅A=c⋅A+k⋅A -
任何一个矩阵 A A A,都存在一个相同形状的矩阵 O O O,满足
A + O = A A + O = A A+O=A
矩阵基本运算代码实现
接之前Matrix类代码,
def __add__(self, another):
assert self.shape() == another.shape()
return Matrix([[a+b for a, b in zip(self.row_vector(i), another.row_vector(i))] for i in range(self.row_num()])
def __sub__(self, another):
assert self.shape() == another.shape()
return Matrix([[a-b for a, b in zip(self.row_vector(i), another.row_vector(i))] for i in range(self.row_num()])
def __mul__(self, k):
return Matrix([[k*a for a in self.row_vector(i)] for i in range(self.row_num()])
def __rmul__(self, k):
return Matrix([[k*a for a in self.row_vector(i)] for i in range(self.row_num()])
def __truediv__(self, k):
return Matrix([[a/k for a in self.row_vector(i)] for i in range(self.row_num()])
def __pos__(self):
return self
def __neg__(self):
return -1 * self
@classmethod
def __zero__(cls, r, c):
return cls([[0] * c] for _ in range(r))