基础线性代数的python实现
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2022-07-12 14:19:43
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1. 行列式
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
%matplotlib inline
import warnings
warnings.filterwarnings('ignore')
行列式的计算 → numpy.linalg.det
计算任何数组a的行列式，这里要求的数组最后两个维度必须是方阵
d = np.array([
[4, 1, 2, 4],
[1, 2, 0, 2],
[10, 5, 2, 0],
[0, 1, 1, 7]
])
print(d)
#计算行列式
np.linalg.det(d)
[[ 4 1 2 4]
[ 1 2 0 2]
[10 5 2 0]
[ 0 1 1 7]]
-1.5099033134902104e-14
def createD(a, x, n):
# 构建函数，生成 n 阶行列式，对角线为x，其余为a
di = np.eye(n)
di = di*x
di[di == 0] = a
return di
d = createD(5, 20, 10)
print(d)
np.linalg.det(d)
[[20. 5. 5. 5. 5. 5. 5. 5. 5. 5.]
[ 5. 20. 5. 5. 5. 5. 5. 5. 5. 5.]
[ 5. 5. 20. 5. 5. 5. 5. 5. 5. 5.]
[ 5. 5. 5. 20. 5. 5. 5. 5. 5. 5.]
[ 5. 5. 5. 5. 20. 5. 5. 5. 5. 5.]
[ 5. 5. 5. 5. 5. 20. 5. 5. 5. 5.]
[ 5. 5. 5. 5. 5. 5. 20. 5. 5. 5.]
[ 5. 5. 5. 5. 5. 5. 5. 20. 5. 5.]
[ 5. 5. 5. 5. 5. 5. 5. 5. 20. 5.]
[ 5. 5. 5. 5. 5. 5. 5. 5. 5. 20.]]
2498818359374.998
2. 矩阵
np.arange(50).reshape(10, 5)
array([[ 0, 1, 2, 3, 4],
[ 5, 6, 7, 8, 9],
[10, 11, 12, 13, 14],
[15, 16, 17, 18, 19],
[20, 21, 22, 23, 24],
[25, 26, 27, 28, 29],
[30, 31, 32, 33, 34],
[35, 36, 37, 38, 39],
[40, 41, 42, 43, 44],
[45, 46, 47, 48, 49]])
print(np.eye(10)*10)
np.eye(5)
[[10. 0. 0. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 10. 0. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 10. 0. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 10. 0. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 10. 0. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 10. 0. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 10. 0. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 10. 0. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 10. 0.]
[ 0. 0. 0. 0. 0. 0. 0. 0. 0. 10.]]
array([[1., 0., 0., 0., 0.],
[0., 1., 0., 0., 0.],
[0., 0., 1., 0., 0.],
[0., 0., 0., 1., 0.],
[0., 0., 0., 0., 1.]])
a = np.arange(10)
print(a)
print(a.shape)
[0 1 2 3 4 5 6 7 8 9]
(10,)
a = np.arange(10).reshape(1, 10)
b = np.arange(10).reshape(10, 1)
print(a)
print(a.shape)
print(b)
print(b.shape)
[[0 1 2 3 4 5 6 7 8 9]]
(1, 10)
[[0]
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]]
(10, 1)
2.2.2 矩阵的运算
ar1 = np.arange(12).reshape(3, 4)
ar2 = np.arange(10, 22).reshape(3, 4)
ar3 = np.ones((3, 4))
ar4 = np.ones((3, 5))
print(ar1)
print(ar2)
print(ar3)
print(ar4)
[[ 0 1 2 3]
[ 4 5 6 7]
[ 8 9 10 11]]
[[10 11 12 13]
[14 15 16 17]
[18 19 20 21]]
[[1. 1. 1. 1.]
[1. 1. 1. 1.]
[1. 1. 1. 1.]]
[[1. 1. 1. 1. 1.]
[1. 1. 1. 1. 1.]
[1. 1. 1. 1. 1.]]
# 矩阵加法
print(ar1 + ar2)
print(ar1 + ar2 + ar3)
[[10 12 14 16]
[18 20 22 24]
[26 28 30 32]]
[[11. 13. 15. 17.]
[19. 21. 23. 25.]
[27. 29. 31. 33.]]
ar1 * 10
array([[ 0, 10, 20, 30],
[ 40, 50, 60, 70],
[ 80, 90, 100, 110]])
# 数组与矩阵相乘
print(ar1 * ar2)
print('--------------------')
# 数组相乘 → numpy里面两个shape相同的数组可以直接相乘，对应位置的值的乘积为结果
# 如果shape不同，则报错
a1 = np.array([2, 3, 4])
b1 = np.array([5, 6, 7]).reshape(3, 1)
# b1 = np.array([5, 6, 7])
c1 = np.dot(a1, b1)
print(a1.shape, b1.shape, c1.shape, c1)
print(c1, type(c1))
a2 = np.array([
[1, 2, 3],
[2, 3, 4]
])
b2 = np.array([
[4, 4],
[5, 5],
[6, 6]
])
c2 = np.dot(a2, b2)
print(a2.shape, b2.shape, c2.shape)
print(c2)
[[ 0 11 24 39]
[ 56 75 96 119]
[144 171 200 231]]
--------------------
(3,) (3, 1) (1,) [56]
[56] <class 'numpy.ndarray'>
(2, 3) (3, 2) (2, 2)
[[32 32]
[47 47]]
# 矩阵乘法
a3 = np.array([
[-2, 4],
[1, -2]
])
b3 = np.array([
[2, 4],
[-3, -6]
])
print(np.dot(a3, b3))
print(np.dot(b3, a3))
[[-16 -32]
[ 8 16]]
[[0 0]
[0 0]]
# 矩阵的转置
A = np.array([
[2, 0, -1],
[1, 3, 2]
])
B = np.array([
[1, 7, -1],
[4, 2, 3],
[2, 0, 1]
])
np.dot(A, B).T
array([[ 0, 17],
[14, 13],
[-3, 10]])
2.2.3 逆矩阵
设A是数域上的一个n阶矩阵，若在相同数域上存在另一个n阶矩阵B，使得： AB=BA=E ，则我们称B是A的逆矩阵，而A则被称为可逆矩阵。注：E为单位矩阵 → 单位矩阵值为1
唯一性：若矩阵A是可逆的，则A的逆矩阵是唯一的
A的逆矩阵的逆矩阵还是A。记作（A-1）-1=A
可逆矩阵A的转置矩阵AT也可逆，并且（AT）-1=（A-1）T (转置的逆等于逆的转置）
两个可逆矩阵的乘积依然可逆
A = np.array([
[1, 2, 3],
[2, 2, 1],
[3, 4, 3]
])
print(A)
print(np.linalg.det(A))
[[1 2 3]
[2 2 1]
[3 4 3]]
1.9999999999999991
# numpy求逆矩阵 B → np.linalg.inv()
B = np.linalg.inv(A)
print(B)
print(np.linalg.det(B))
[[ 1. 3. -2. ]
[-1.5 -3. 2.5]
[ 1. 1. -1. ]]
0.5000000000000004
E = np.dot(A, B)
print(E)
print(np.linalg.det(E))
[[ 1.00000000e+00 2.22044605e-16 0.00000000e+00]
[-2.22044605e-16 1.00000000e+00 4.44089210e-16]
[-2.22044605e-16 2.22044605e-16 1.00000000e+00]]
1.0
A_bs = B*np.linalg.det(A)
print(A_bs)
print(np.linalg.det(A_bs))
[[ 2. 6. -4.]
[-3. -6. 5.]
[ 2. 2. -2.]]
3.9999999999999947
a = np.array([
[4, 1, 2, 4],
[1, 2, 0, 2],
[1, 0, 2, 0],
[0, 1, 1, 0]
])
print(np.linalg.inv(a))
np.linalg.det(a)
[[-2. 4. 5. -6. ]
[-1. 2. 2. -2. ]
[ 1. -2. -2. 3. ]
[ 2. -3.5 -4.5 5. ]]
1.9999999999999993
a = np.array([
[1,1,1],
[1,2,4],
[1,3,9]
])
a1 = np.array([
[2,1,1],
[3,2,4],
[5,3,9]
])
a2 = np.array([
[1,2,1],
[1,3,4],
[1,5,9]
])
a3 = np.array([
[1,1,2],
[1,2,3],
[1,3,5]
])
x1 = np.linalg.det(a1)/np.linalg.det(a)
x2 = np.linalg.det(a2)/np.linalg.det(a)
x3 = np.linalg.det(a3)/np.linalg.det(a)
print('该方程的三个根为：x1=%.2f, x2=%.2f, x3=%.2f' % (x1,x2,x3))
该方程的三个根为：x1=2.00, x2=-0.50, x3=0.50
2.3.2 线性方程组
# 计算秩
a = np.array([
[1,2,3],
[2,3,-5],
[4,7,1]
])
b = np.array([
[3,2,0,5,0],
[3,-2,3,6,-1],
[2,0,1,5,-3],
[1,6,-4,-1,4]
])
r1 = np.linalg.matrix_rank(a)
r2 = np.linalg.matrix_rank(b)
print(r1, r2)
2 3
2.4.3 向量的性质
- np.linalg.eigvals() → 计算矩阵的特征值
- np.linalg.eig() → 返回包含特征值和对应特征向量的元组
a = np.array([
[3, -1],
[-1, 3]
])
print(np.linalg.eigvals(a))
print(np.linalg.eig(a))
[4. 2.]
(array([4., 2.]), array([[ 0.70710678, 0.70710678],
[-0.70710678, 0.70710678]]))
2.4.4 矩阵的对角化
- np.diag() → 对角化
a = np.array([
[-2, 1, 1],
[0, 2, 0],
[-4, 1, 3]
])
print(np.linalg.eigvals(a))
np.diag(np.linalg.eigvals(a))
[-1. 2. 2.]
array([[-1., 0., 0.],
[ 0., 2., 0.],
[ 0., 0., 2.]])
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