线性代数-Python-03：矩阵的变换 - 手写Matrix Transformation及numpy中的用法

一、代码仓库

https://github.com/Chufeng-Jiang/Python-Linear-Algebra-for-Beginner/tree/main

二、旋转矩阵的推导及图形学中的矩阵变换

2.1 让横坐标扩大a倍，纵坐标扩大b倍

2.2 关于x轴翻转

2.3 关于y轴翻转

2.4 关于原点翻转（x轴，y轴均翻转）

2.5 沿x方向错切

2.6 沿y方向错切

2.7 旋转

theta = math.pi / 3
T = Matrix([[math.cos(theta), math.sin(theta)], 
	        [-math.sin(theta), math.cos(theta)]])

2.8 单位矩阵

2.9 矩阵的逆

三、看待矩阵的关键视角：用矩阵表示空间

3.1 行视角

3.2 列视角

3.3 标准单位向量和列视角

3.4 矩阵表示空间

四、代码

matrix.py

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)])

    @classmethod
    def identity(cls, n):
        """返回一个n行n列的单位矩阵"""
        m = [[0]*n for _ in range(n)]
        for i in range(n):
            m[i][i] = 1;
        return cls(m)

    def T(self):
        """返回矩阵的转置矩阵"""
        return Matrix([[e for e in self.col_vector(i)]
                       for i in range(self.col_num())])

    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 dot(self, another):
        """返回矩阵乘法的结果"""
        if isinstance(another, Vector):
            # 矩阵和向量的乘法
            assert self.col_num() == len(another), \
                "Error in Matrix-Vector Multiplication."
            return Vector([self.row_vector(i).dot(another) for i in range(self.row_num())])

        if isinstance(another, Matrix):
            # 矩阵和矩阵的乘法
            assert self.col_num() == another.row_num(), \
                "Error in Matrix-Matrix Multiplication."
            return Matrix([[self.row_vector(i).dot(another.col_vector(j)) for j in range(another.col_num())]
                           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__

matrix_transformation

import matplotlib.pyplot as plt
from playLA.Matrix import Matrix
from playLA.Vector import Vector
import math


if __name__ == "__main__":

    points = [[0, 0], [0, 5], [3, 5], [3, 4], [1, 4], [1, 3], [2, 3], [2, 2], [1, 2], [1, 0]]
    x = [point[0] for point in points]
    y = [point[1] for point in points]

    plt.figure(figsize=(5, 5))
    plt.xlim(-10, 10)
    plt.ylim(-10, 10)
    plt.plot(x, y)
    # plt.show()

    P = Matrix(points)

    # T = Matrix([[2, 0], [0, 1.5]])
    # T = Matrix([[1, 0], [0, -1]])
    # T = Matrix([[-1, 0], [0, 1]])
    # T = Matrix([[-1, 0], [0, -1]])
    # T = Matrix([[1, 0.5], [0, 1]])
    # T = Matrix([[1, 0], [0.5, 1]])

    theta = math.pi / 3
    T = Matrix([[math.cos(theta), math.sin(theta)], [-math.sin(theta), math.cos(theta)]])

    # 逆时针旋转90度
    # theta = math.pi / -2
    # T = Matrix([[math.cos(theta), math.sin(theta)], [-math.sin(theta), math.cos(theta)]])

    # 根据矩阵表示空间的法则，直接写出的逆时针旋转90度的变换矩阵
    #T = Matrix([[0, -1], [1, 0]])

    P2 = T.dot(P.T())
    plt.plot([P2.col_vector(i)[0] for i in range(P2.col_num())],
             [P2.col_vector(i)[1] for i in range(P2.col_num())])
    plt.show()

numpy_matrix.py

import numpy as np

if __name__ == "__main__":

    # 矩阵的创建
    A = np.array([[1, 2], [3, 4]])
    print(A)

    # 矩阵的属性
    print(A.shape)
    print(A.T)

    # 获取矩阵的元素
    print(A[1, 1])
    print(A[0])
    print(A[:, 0])
    print(A[1, :])

    # 矩阵的基本运算
    B = np.array([[5, 6], [7, 8]])
    print(A + B)
    print(A - B)
    print(10 * A)
    print(A * 10)
    print(A * B)
    print(A.dot(B))

    p = np.array([10, 100])
    print(A + p)
    print(A + 1)

    print(A.dot(p))

    # 单位矩阵
    I = np.identity(2)
    print(I)
    print(A.dot(I))
    print(I.dot(A))

    # 逆矩阵
    invA = np.linalg.inv(A)
    print(invA)
    print(invA.dot(A))
    print(A.dot(invA))

    # C = np.array([[1, 2, 3], [4, 5, 6]])
    # np.linalg.inv(C)