线性代数(3)——矩阵基础

博客介绍了矩阵是对向量的拓展,有行向量和列向量两个视角,行数和列数相等时为方阵。详细阐述了矩阵的基本运算,包括加法、数乘,以及矩阵运算的性质,如交换律、结合律等,还提及了矩阵基本运算的代码实现。

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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,
A = ( a 11 a 12 . . . a 1 c a 21 a 22 . . . a 2 c . . . . . . . . . . . . a r 1 a r 2 . . . a r c ) B = ( b 11 b 12 . . . b 1 c b 21 b 22 . . . b 2 c . . . . . . . . . . . . b r 1 b r 2 . . . b r c ) A=\begin{pmatrix}a_{11}&amp;a_{12}&amp;...&amp;a_{1c}\\a_{21}&amp;a_{22}&amp;...&amp;a_{2c}\\...&amp;...&amp;...&amp;...\\a_{r1}&amp;a_{r2}&amp;...&amp;a_{rc}\end{pmatrix}B=\begin{pmatrix}b_{11}&amp;b_{12}&amp;...&amp;b_{1c}\\b_{21}&amp;b_{22}&amp;...&amp;b_{2c}\\...&amp;...&amp;...&amp;...\\b_{r1}&amp;b_{r2}&amp;...&amp;b_{rc}\end{pmatrix} A=a11a21...ar1a12a22...ar2............a1ca2c...arcB=b11b21...br1b12b22...br2............b1cb2c...brc
A + B = ( a 11 + b 11 a 12 + b 12 . . . a 1 c + b 1 c a 21 + b 21 a 22 + b 22 . . . a 2 c + b 2 c . . . . . . . . . . . . a r 1 + b r 1 a r 2 + b r 2 . . . a r c + b r c ) A+B=\begin{pmatrix}a_{11}+b_{11}&amp;a_{12}+b_{12}&amp;...&amp;a_{1c}+b_{1c}\\a_{21}+b_{21}&amp;a_{22}+b_{22}&amp;...&amp;a_{2c}+b_{2c}\\...&amp;...&amp;...&amp;...\\a_{r1}+b_{r1}&amp;a_{r2}+b_{r2}&amp;...&amp;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,
A = ( a 11 a 12 . . . a 1 c a 21 a 22 . . . a 2 c . . . . . . . . . . . . a r 1 a r 2 . . . a r c ) A=\begin{pmatrix}a_{11}&amp;a_{12}&amp;...&amp;a_{1c}\\a_{21}&amp;a_{22}&amp;...&amp;a_{2c}\\...&amp;...&amp;...&amp;...\\a_{r1}&amp;a_{r2}&amp;...&amp;a_{rc}\end{pmatrix} A=a11a21...ar1a12a22...ar2............a1ca2c...arc
k ⋅ A = ( k ⋅ a 11 k ⋅ a 12 . . . k ⋅ a 1 c k ⋅ a 21 k ⋅ a 22 . . . k ⋅ a 2 c . . . . . . . . . . . . k ⋅ a r 1 k ⋅ a r 2 . . . k ⋅ a r c ) k\cdot A=\begin{pmatrix}k\cdot a_{11}&amp;k\cdot a_{12}&amp;...&amp;k\cdot a_{1c}\\k\cdot a_{21}&amp;k\cdot a_{22}&amp;...&amp;k\cdot a_{2c}\\...&amp;...&amp;...&amp;...\\k\cdot a_{r1}&amp;k\cdot a_{r2}&amp;...&amp;k\cdot a_{rc}\end{pmatrix} kA=ka11ka21...kar1ka12ka22...kar2............ka1cka2c...karc

矩阵运算性质

  1. 交换律
    A + B = B + A A + B = B + A A+B=B+A

  2. 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) (ck)A=c(kA) 。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)=kA+kB
    ( c + k ) ⋅ A = c ⋅ A + k ⋅ A (c + k) \cdot A = c\cdot A + k\cdot A (c+k)A=cA+kA

  3. 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))
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