Linear Algebra Study Note
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Wrote on May 25, 2020, 10:06 p.m.
- Vector 向量
- Matrix 矩阵
- Linear System 线性系统
- Inverse of Matrix 矩阵的逆
- Elementary Matrix 初等矩阵
- Matrix Decomposition 矩阵的分解
- Linear Combination 线性组合
- Linear Space 线性空间
- Orthonormal Basis 正交基和标准正交基
- Coordinate Transformation 坐标转换
- Determinant 行列式
- Eigenvalues and Eigenvectors 特征值和特征向量
- Singular Value Decomposition 奇异值分解
Customized Linear Algebra Calculation Module
# _global.py
# -*- coding: utf-8 -*-
"""
Created by @南卡统爷 on 5/19/20 1:41 PM.
"""
__author__ = '@JiayuWang'
EPSILON = 1e-8
def is_zero(x):
return abs(x) < EPSILON
def is_equal(a, b):
return abs(a - b) < EPSILON
# Vector.py
# -*- coding: utf-8 -*-
"""
Created by @南卡统爷 on 5/18/20 9:55 PM.
"""
__author__ = '@JiayuWang'
import math
from ._global import is_zero, is_equal
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 norm(self):
"""返回向量的模（长度）"""
return math.sqrt(sum(e ** 2 for e in self))
def normalize(self):
"""返回向量的单位向量"""
if is_zero(self.norm()):
raise ZeroDivisionError("Normalize error! Norm is zero.")
return 1 / self.norm() * Vector(self) # 也可以写成 Vect(self._values) 但因为定义了 __iter__ 所以可以直接用 self
# return Vector(self._values) / self.norm()
def underlying_list(self):
"""返回向量的底层列表"""
return self._values[:]
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 equal"
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 adding. Length of vectors must be equal"
return Vector([a - b for a, b in zip(self, another)])
def dot(self, another):
"""向量点乘，返回结果标量"""
assert len(self) == len(another), \
"Error in adding. Length of vectors must be equal"
return sum(a * b for a, b in zip(self, another))
def __mul__(self, k):
"""向量乘法，返回结果向量: self * k """
return Vector([k * e for e in self])
def __rmul__(self, k):
"""向量乘法，返回结果向量: 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 __eq__(self, other):
"""返回向量是否相等"""
other_list = other.underlying_list()
if(len(other_list) != len(self._values)):
return False
return all(is_equal(x, y) for x, y in zip(self._values, other_list))
def __neq__(self, other):
"""返回向量是否不等"""
return not(self == other)
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))
# Matrix.py
# -*- coding: utf-8 -*-
__author__ = '@JiayuWang'
from .Vector import Vector
class Matrix:
def __init__(self, list2d):
if isinstance(list2d[0], list):
self._values = [row[:] for row in list2d]
elif isinstance(list2d[0], Vector):
self._values = [row.underlying_list() 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列的零矩阵"""
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):
"""返回两个矩阵的相加结果"""
assert self.shape() == another.shape(), \
"Error in adding. Shape of matrix must be the 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):
"""返回两个矩阵的相减结果"""
assert self.shape() == another.shape(), \
"Error in adding. Shape of matrix must be the 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):
"""返回矩阵的数量乘法: self * k"""
return Matrix([[e*k for e in self.row_vector(i)] for i in range(self.row_num())])
def __rmul__(self, k):
"""返回矩阵的数量乘法: k * self"""
return self * k
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 __truediv__(self, k):
"""返回矩阵的数量除法: self / k"""
return (1/k) * self
def __pos__(self):
"""返回矩阵的取正结果"""
return 1*self
def __neg__(self):
"""返回矩阵的取负结果"""
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 shape(self):
"""返回矩阵的形状：（行数，列数）"""
return len(self._values), len(self._values[0])
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 __repr__(self):
return "Matrix({})".format(self._values)
__str__ = __repr__
# LU.py
# -*- coding: utf-8 -*-
"""
Created by @南卡统爷 on 5/24/20 3:52 PM.
"""
__author__ = '@JiayuWang'
from .Matrix import Matrix
from .Vector import Vector
from ._global import is_zero
def lu(matrix):
assert matrix.row_num() == matrix.col_num(), "Matrix must be a square matrix"
n = matrix.row_num()
A = [matrix.row_vector(i) for i in range(n)]
L = [[1.0 if i == j else 0.0 for i in range(n)] for j in range(n)]
for i in range(n):
# 看A[i][i]位置是否可以是主元
if is_zero(A[i][i]):
return None, None
else:
for j in range(i+1, n):
p = A[j][i] / A[i][i]
A[j] = A[j] - p * A[i]
L[j][i] = p
return Matrix(L), Matrix([A[i].underlying_list() for i in range(n)])
# LinearSystem.py
# -*- coding: utf-8 -*-
"""
Created by @南卡统爷 on 5/23/20 1:19 AM.
"""
__author__ = '@JiayuWang'
from .Matrix import Matrix
from .Vector import Vector
from ._global import is_zero
class LinearSystem:
def __init__(self, A, b=None):
"""A是增广矩阵左侧的系数矩阵，b是右侧的结果的列向量"""
assert b is None or A.row_num() == len(b), "row number of A must be equal to the length of b"
self._m = A.row_num()
self._n = A.col_num()
if b is None:
self.Ab = [A.row_vector(i) for i in range(self._m)]
if isinstance(b, Vector):
self.Ab = [Vector(A.row_vector(i).underlying_list() + [b[i]]) for i in range(self._m)]
if isinstance(b, Matrix):
self.Ab = [Vector(A.row_vector(i).underlying_list() + b.row_vector(i).underlying_list()) for i in
range(self._m)]
self.pivots = []
def _max_row(self, index_i, index_j, n):
"""找出指定列的数字最大的那行的行号"""
best, ret = self.Ab[index_i][index_j], index_i
for i in range(index_i + 1, n):
if self.Ab[i][index_j] > best:
best, ret = self.Ab[i][index_j], i
return ret
def _forward(self):
i, k = 0, 0
while i < self._m and k < self._n:
# 看Ab[i][i]位置是否可以是主元
max_row = self._max_row(i, k, self._m)
self.Ab[i], self.Ab[max_row] = self.Ab[max_row], self.Ab[i]
if is_zero(self.Ab[i][k]):
k += 1
else:
# 将主元归为一
self.Ab[i] = self.Ab[i] / self.Ab[i][k]
for j in range(i + 1, self._m):
self.Ab[j] = self.Ab[j] - self.Ab[j][k] * self.Ab[i]
self.pivots.append(k)
i += 1
def _backward(self):
n = len(self.pivots)
for i in range(n - 1, -1, -1):
k = self.pivots[i]
# Ab[i][k]为主元
for j in range(i - 1, -1, -1):
self.Ab[j] = self.Ab[j] - self.Ab[j][i] * self.Ab[i]
def gauss_jordan_elimination(self):
"""如果有解，返回True；如果没有解，返回False"""
self._forward()
self._backward()
for i in range(len(self.pivots), self._m):
if not is_zero(self.Ab[i][-1]):
return False
return True
def fancy_print(self):
for i in range(self._m):
print(" ".join(str(self.Ab[i][j]) for j in range(self._n)), end=" ")
print("|", self.Ab[i][-1])
def inv(A):
if A.row_num() != A.col_num():
return None
n = A.row_num()
ls = LinearSystem(A, Matrix.identity(n))
if not ls.gauss_jordan_elimination():
return None
invA = [[row[i] for i in range(n, 2*n)] for row in ls.Ab]
return Matrix(invA)
def rank(A):
"""求矩阵的秩"""
ls = LinearSystem(A)
ls.gauss_jordan_elimination()
zero = Vector.zero(A.col_num())
return sum([row != zero for row in ls.Ab])
# GramSchmidtProcess.py
# -*- coding: utf-8 -*-
"""
Created by @南卡统爷 on 5/26/20 1:29 AM.
"""
__author__ = '@JiayuWang'
from .Vector import Vector
from .Matrix import Matrix
from .LinearSystem import rank
def gram_schmidt_process(basis):
#把所有的向量按照行的方式排列在一起然后高斯消元，只要结果没有零行就说明线性无关
matrix = Matrix(basis)
assert rank(matrix) == len(basis)
res = [basis[0]]
for i in range(1, len(basis)):
p = basis[i]
for r in res:
p = p - basis[i].dot(r) / r.dot(r) * r
res.append(p)
return res
def qr(A):
assert A.row_num() == A.col_num(), "A must be square"
basis = [A.col_vector(i) for i in range(A.col_num())]
P = gram_schmidt_process(basis)
# 基于标准正交基
Q = Matrix([v/v.norm() for v in P]).T()
R = Q.T().dot(A)
return Q, R
