Numpy_matrix

import numpy as np

矩阵生成

x = np.matrix([[1,2,3],[4,5,6]])
y = np.matrix([1,2,3,4,5,6])

print(x,y,x[0,0],x[0])  #数组中 x[1][1] 和 x[1,1]  表示同一个元素 矩阵不是

[[1 2 3]
[4 5 6]] [[1 2 3 4 5 6]] 1 [[1 2 3]]

矩阵转置

x = np.mat([[1,2,3],[4,5,6]])
y = np.mat([1,2,3,4,5,6])

x.T

matrix([[1, 4],
[2, 5],
[3, 6]])

y.T

matrix([[1],
[2],
[3],
[4],
[5],
[6]])

matrix([[1, 2, 3],
[4, 5, 6]])

x.mean() #所有元素平均值

3.5

x.mean(axis=0) #纵向平均值

matrix([[2.5, 3.5, 4.5]])

x.mean(axis=0).shape #数组形状

(1, 3)

x.mean(axis=1) #横向平均值

matrix([[2.],
[5.]])

x.max(axis=1) #横向最大值

matrix([[3],
[6]])

x.diagonal() #对角线元素

matrix([[1, 5]])

x.nonzero() #非零元素下标 分别返回行下标，列下标

(array([0, 0, 0, 1, 1, 1], dtype=int64),
array([0, 1, 2, 0, 1, 2], dtype=int64))

矩阵乘法

x = np.matrix([[1,2,3],[4,5,6]])
y = np.matrix([[1,2],[3,4],[5,6]])

print(x)

[[1 2 3]
[4 5 6]]

print(y)

[[1 2]
[3 4]
[5 6]]

x*y

matrix([[22, 28],
[49, 64]])

计算相关系数矩阵

np.corrcoef([1,2,3,4],[4,3,2,1]) #负相关 变化方向相反

array([[ 1., -1.],
[-1., 1.]])

np.corrcoef([1,2,3,4],[8,3,2,1]) #负相关 变化方向相反

array([[ 1. , -0.91350028],
[-0.91350028, 1. ]])

np.corrcoef([1,2,3,4],[1,2,3,4]) #正相关 变化方向相同

array([[1., 1.],
[1., 1.]])

np.corrcoef([1,2,3,4],[1,2,3,40]) #正相关变化趋势接近

array([[1. , 0.8010362],
[0.8010362, 1. ]])

方差协方差标准差

np.cov([1,1,1,1,1]) #方差

array(0.)

np.std([1,1,1,1,1]) #标准差

0.0

x = [-2.1,-1,4.3]
y = [3,1.1,0.12]
X = np.vstack((x,y)) #垂直堆叠矩阵

array([[-2.1 , -1. , 4.3 ],
[ 3. , 1.1 , 0.12]])

np.cov(X) #协方差

array([[11.71 , -4.286 ],
[-4.286 , 2.14413333]])

np.cov(x,y)

array([[11.71 , -4.286 ],
[-4.286 , 2.14413333]])

np.std(x) #标准差

2.794041278626117

np.std(X,axis=1)

array([2.79404128, 1.19558447])

np.cov(x) #方差

array(11.71)

特征值与特征向量

A = np.mat([[1,-3,3],[3,-5,3],[6,-6,4]])
e,v =np.linalg.eig(A) #特征向量与特征值

array([ 4.+0.00000000e+00j, -2.+1.10465796e-15j, -2.-1.10465796e-15j])

matrix([[-0.40824829+0.j , 0.24400118-0.40702229j,
0.24400118+0.40702229j],
[-0.40824829+0.j , -0.41621909-0.40702229j,
-0.41621909+0.40702229j],
[-0.81649658+0.j , -0.66022027+0.j ,
-0.66022027-0.j ]])

np.dot(A,v) #矩阵和特征向量乘积

matrix([[-1.63299316+0.0000000e+00j, -0.48800237+8.1404458e-01j,
-0.48800237-8.1404458e-01j],
[-1.63299316+0.0000000e+00j, 0.83243817+8.1404458e-01j,
0.83243817-8.1404458e-01j],
[-3.26598632+0.0000000e+00j, 1.32044054-4.4408921e-16j,
1.32044054+4.4408921e-16j]])

np.isclose(np.dot(A,v),e*v) #验证二者是否相等

matrix([[False, False, False],
[False, False, False],
[False, False, False]])

np.linalg.det(A-np.eye(3,3)*e)  #det 计算行列式的函数 |A-（蓝不大）*E|

(-3.660807638291249e-30+5.965152994198125e-14j)

计算逆矩阵

x = np.matrix([[1,2,3],[4,5,6],[7,8,0]])
y = np.linalg.inv(x)

matrix([[-1.77777778, 0.88888889, -0.11111111],
[ 1.55555556, -0.77777778, 0.22222222],
[-0.11111111, 0.22222222, -0.11111111]])

x*y #对角线元素为 1 其他元素为0或近似为0

matrix([[ 1.00000000e+00, 1.11022302e-16, 0.00000000e+00],
[-1.11022302e-15, 1.00000000e+00, 0.00000000e+00],
[ 0.00000000e+00, 0.00000000e+00, 1.00000000e+00]])

y*x

matrix([[ 1.00000000e+00, -4.44089210e-16, 0.00000000e+00],
[ 2.22044605e-16, 1.00000000e+00, 0.00000000e+00],
[ 1.11022302e-16, 1.11022302e-16, 1.00000000e+00]])

求解线性方程组

a = np.array([[3,1],[1,2]]) #系数矩阵
b = np.array([9,8])         #系数矩阵
x = np.linalg.solve(a,b)  #求解
x

array([2., 3.])

np.dot(a,x) #验证

array([9., 8.])

np.linalg.lstsq(a,b) #最小二乘解  返回解余项 a的秩 a的奇异值

:1: FutureWarning: rcond parameter will change to the default of machine precision times max(M, N) where M and N are the input matrix dimensions.
To use the future default and silence this warning we advise to pass rcond=None, to keep using the old, explicitly pass rcond=-1.
np.linalg.lstsq(a,b) #最小二乘解返回解余项 a的秩 a的奇异值

(array([2., 3.]), array([], dtype=float64), 2, array([3.61803399, 1.38196601]))

计算向量和矩阵的范数

x = np.matrix([[1,2],[3,-4]])
np.linalg.norm(x)

5.477225575051661

np.linalg.norm(x,-2)

1.9543950758485487

np.linalg.norm([1,2,0,3,4],2)

5.477225575051661

奇异值分解

a = np.matrix([[1,2,3],[4,5,6],[7,8,9]])
u,s,v = np.linalg.svd(a) #奇异值分解
u

matrix([[-0.21483724, 0.88723069, 0.40824829],
[-0.52058739, 0.24964395, -0.81649658],
[-0.82633754, -0.38794278, 0.40824829]])

array([1.68481034e+01, 1.06836951e+00, 4.41842475e-16])

matrix([[-0.47967118, -0.57236779, -0.66506441],
[-0.77669099, -0.07568647, 0.62531805],
[-0.40824829, 0.81649658, -0.40824829]])

u*np.diag(s)*v #验证

matrix([[1., 2., 3.],
[4., 5., 6.],
[7., 8., 9.]])

函数向量化

mat = np.matrix([[1,2,3],[4,5,6]])
mat