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python 矩阵除法_Python线性代数学习笔记——矩阵的基本运算和基本性质,实现矩阵的基本运算...

矩阵除法基本性质

当学习完矩阵的定义以后,我们来学习矩阵的基本运算,与基本性质

矩阵的基本运算:矩阵的加法,每一个对应元素相加,对应结果的矩阵

例子:矩阵A和矩阵B表示的是同学上学期和下学期的课程的成绩,两个矩阵相加就表示一学年科目成绩的总和

矩阵的数量乘法:一个数乘于一个矩阵

还是接着上面学生成绩的例子:

矩阵数量乘法可以理解为,求两学期学生科目成绩的平均分1/2(A+B),因为之前我们已经算出了一学年科目的成绩总和,现在只需要乘于二分之一就可以了。

矩阵的数量乘法还有一个几何的直观理解:

下图的矩阵P可以理解为3个行向量组成,这3个行向量表示的是二维平面坐标系中的一个点,就是表示一个三角形,矩阵的数量乘法2.P之后,这个三角形就缩放变大了

矩阵的基本运算性质

简单证明:k ⋅(A + B) = k ⋅ A + k ⋅ B(这都还用证????不过出于数学逻辑思维的严谨,还是需要证明的)

两个矩阵:

实现矩阵的基本运算

之前定义的向量类Vector:

import math

from ._globals import EPSILON

class Vector:

def __init__(self, lst):

self._values = list(lst)

@classmethod

def zero(cls, dim):

"""返回一个dim维的零向量"""

return cls([0] * dim)

def __add__(self, another):

"""向量加法,返回结果向量"""

assert len(self) == len(another), \

"Error in adding. Length of vectors must be same."

return Vector([a + b for a, b in zip(self, another)])

def __sub__(self, another):

"""向量减法,返回结果向量"""

assert len(self) == len(another), \

"Error in subtracting. Length of vectors must be same."

return Vector([a - b for a, b in zip(self, another)])

def norm(self):

"""返回向量的模"""

return math.sqrt(sum(e**2 for e in self))

def normalize(self):

"""返回向量的单位向量"""

if self.norm() < EPSILON:

raise ZeroDivisionError("Normalize error! norm is zero.")

return Vector(self._values) / self.norm()

def dot(self, another):

"""向量点乘,返回结果标量"""

assert len(self) == len(another), \

"Error in dot product. Length of vectors must be same."

return sum(a * b for a, b in zip(self, another))

def __mul__(self, k):

"""返回数量乘法的结果向量:self * k"""

return Vector([k * e for e in self])

def __rmul__(self, k):

"""返回数量乘法的结果向量:k * self"""

return self * k

def __truediv__(self, k):

"""返回数量除法的结果向量:self / k"""

return (1 / k) * self

def __pos__(self):

"""返回向量取正的结果向量"""

return 1 * self

def __neg__(self):

"""返回向量取负的结果向量"""

return -1 * self

def __iter__(self):

"""返回向量的迭代器"""

return self._values.__iter__()

def __getitem__(self, index):

"""取向量的第index个元素"""

return self._values[index]

def __len__(self):

"""返回向量长度(有多少个元素)"""

return len(self._values)

def __repr__(self):

return "Vector({})".format(self._values)

def __str__(self):

return "({})".format(", ".join(str(e) for e in self._values))

定义一个内部使用的文件_globals,用来存储全局使用的变量 EPSILON,用来判断精度用的

EPSILON = 1e-8

定义的矩阵类Matrix:

from .Vector import Vector

class Matrix:

def __init__(self, list2d):

self._values = [row[:] for row in list2d]

@classmethod

def zero(cls, r, c):

"""返回一个r行c列的零矩阵"""

return cls([[0] * c for _ in range(r)])

def __add__(self, another):

"""返回两个矩阵的加法结果"""

assert self.shape() == another.shape(), \

"Error in adding. Shape of matrix must be same."

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):

"""返回两个矩阵的减法结果"""

assert self.shape() == another.shape(), \

"Error in subtracting. Shape of matrix must be same."

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):

"""返回矩阵的数量乘结果: self * k"""

return Matrix([[e * k for e in self.row_vector(i)]

for i in range(self.row_num())])

def __rmul__(self, k):

"""返回矩阵的数量乘结果: k * self"""

return self * k

def __truediv__(self, k):

"""返回数量除法的结果矩阵:self / k"""

return (1 / k) * self

def __pos__(self):

"""返回矩阵取正的结果"""

return 1 * self

def __neg__(self):

"""返回矩阵取负的结果"""

return -1 * self

def row_vector(self, index):

"""返回矩阵的第index个行向量"""

return Vector(self._values[index])

def col_vector(self, index):

"""返回矩阵的第index个列向量"""

return Vector([row[index] for row in self._values])

def __getitem__(self, pos):

"""返回矩阵pos位置的元素"""

r, c = pos

return self._values[r][c]

def size(self):

"""返回矩阵的元素个数"""

r, c = self.shape()

return r * c

def row_num(self):

"""返回矩阵的行数"""

return self.shape()[0]

__len__ = row_num

def col_num(self):

"""返回矩阵的列数"""

return self.shape()[1]

def shape(self):

"""返回矩阵的形状: (行数, 列数)"""

return len(self._values), len(self._values[0])

def __repr__(self):

return "Matrix({})".format(self._values)

__str__ = __repr__

测试代码:

from playLA.Matrix import Matrix

if __name__ == "__main__":

matrix = Matrix([[1, 2], [3, 4]])

print(matrix)

print("matrix.shape = {}".format(matrix.shape()))

print("matrix.size = {}".format(matrix.size()))

print("len(matrix) = {}".format(len(matrix)))

print("matrix[0][0] = {}".format(matrix[0, 0]))

matrix2 = Matrix([[5, 6], [7, 8]])

print(matrix2)

print("add: {}".format(matrix + matrix2))

print("subtract: {}".format(matrix - matrix2))

print("scalar-mul: {}".format(2 * matrix))

print("scalar-mul: {}".format(matrix * 2))

print("zero_2_3: {}".format(Matrix.zero(2, 3)))

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