线性代数

In [138]: x = np.array([[1., 2., 3.], [4., 5., 6.]])

In [139]: y = np.array([[6., 23.], [-1., 7], [8., 9.]])

In [140]: x
Out[140]:
array([[1., 2., 3.],
       [4., 5., 6.]])

In [141]: y
Out[141]:
array([[ 6., 23.],
       [-1.,  7.],
       [ 8.,  9.]])

In [142]: x.dot(y)
Out[142]:
array([[ 28.,  64.],
       [ 67., 181.]])

x.dot(y) 等价np.dot(x, y)

In [143]: np.dot(x, y)
Out[143]:
array([[ 28.,  64.],
       [ 67., 181.]])

特殊符号@也作为中缀操作符 用于点乘矩阵操作

In [145]: x @ np.ones(3)
Out[145]: array([ 6., 15.])

numpy.linalg拥有一个矩阵分解的标准函数集，以及其他常用函数，例如求逆和行列式求解。

In [146]: from numpy.linalg import inv, qr

In [147]: X = np.random.randn(5, 5)

In [148]: mat = X .T .dot(X)

In [149]: inv(mat)
Out[149]:
array([[  3.81634213,  -0.33734926,  -8.29257805, -19.34806895,
         12.16184347],
       [ -0.33734926,   0.24647885,   1.26970515,   2.94427259,
         -1.97488616],
       [ -8.29257805,   1.26970515,  26.08526755,  60.89076132,
        -38.32595465],
       [-19.34806895,   2.94427259,  60.89076132, 142.58472207,
        -89.65423579],
       [ 12.16184347,  -1.97488616, -38.32595465, -89.65423579,
         56.58562833]])

In [150]: mat.dot(inv(mat))
Out[150]:
array([[ 1.00000000e+00,  0.00000000e+00, -8.88178420e-16,
         0.00000000e+00,  0.00000000e+00],
       [ 0.00000000e+00,  1.00000000e+00,  0.00000000e+00,
        -5.68434189e-14,  5.68434189e-14],
       [ 1.42108547e-14,  0.00000000e+00,  1.00000000e+00,
        -5.68434189e-14,  1.70530257e-13],
       [-7.10542736e-15,  0.00000000e+00, -2.84217094e-14,
         1.00000000e+00, -5.68434189e-14],
       [-1.42108547e-14,  0.00000000e+00,  5.68434189e-14,
         0.00000000e+00,  1.00000000e+00]])

In [151]: q,r = qr(mat)

In [152]: q
Out[152]:
array([[-0.83864776, -0.39066785,  0.3558009 ,  0.07715658,  0.10724336],
       [ 0.41520162, -0.71433003,  0.07867346,  0.55753694, -0.01741458],
       [-0.2870064 , -0.25961934, -0.85786848, -0.00839819, -0.33795897],
       [ 0.10910234, -0.22246167,  0.34657706, -0.43987118, -0.79057269],
       [ 0.17320954, -0.46927586, -0.1056514 , -0.69974441,  0.4989731 ]])

In [153]: r
Out[153]:
array([[-1.03628672e+00,  5.07388454e+00, -3.59320544e+00,
         3.42845808e+00,  3.40120764e+00],
       [ 0.00000000e+00, -9.79075704e+00, -5.06108475e+00,
        -3.59439551e+00, -9.47287516e+00],
       [ 0.00000000e+00,  0.00000000e+00, -1.31905677e+01,
         4.44252096e+00, -1.89722959e+00],
       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,
        -2.88582436e+00, -4.58466452e+00],
       [ 0.00000000e+00,  0.00000000e+00,  0.00000000e+00,
         0.00000000e+00,  8.81801825e-03]])

函数	描述
diag	将一个方阵的对角（或非对角）元素作为一维数组返回，或者将一维数组转换成一个方阵，并且在非对角线上有零点
dot	矩阵点乘
trace	计算对角元素和
det	计算矩阵的行列式
eig	计算方阵的特征值和特征向量
inv	计算矩阵的逆矩阵
pinv	计算矩阵的Moore-Penrose伪逆
qr	计算QR分解
svd	计算奇异值分解（SVD）
solve	求解x的线性系统Ax = b，其中A是方阵
lstsq	计算Ax = b的最小二乘解