numpy向量计算

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向量与矩阵的计算

【a】向量内积：dot

$\rm \mathbf{a}\cdot\mathbf{b} = \sum_ia_ib_i$

a = np.array([1,2,3])
b = np.array([1,3,5])
a.dot(b)

22

【b】向量范数和矩阵范数：np.linalg.norm

在矩阵范数的计算中，最重要的是ord参数，可选值如下：

ord	norm for matrices	norm for vectors
None	Frobenius norm	2-norm
'fro'	Frobenius norm	/
'nuc'	nuclear norm	/
inf	max(sum(abs(x), axis=1))	max(abs(x))
-inf	min(sum(abs(x), axis=1))	min(abs(x))
0	/	sum(x != 0)
1	max(sum(abs(x), axis=0))	as below
-1	min(sum(abs(x), axis=0))	as below
2	2-norm (largest sing. value)	as below
-2	smallest singular value	as below
other	/	sum(abs(x)ord)(1./ord)

matrix_target =  np.arange(4).reshape(-1,2)
matrix_target

array([[0, 1],
       [2, 3]])

np.linalg.norm(matrix_target, 'fro')

3.7416573867739413

np.linalg.norm(matrix_target, np.inf)

5.0

np.linalg.norm(matrix_target, 2)

3.702459173643833

vector_target =  np.arange(4)
vector_target

array([0, 1, 2, 3])

np.linalg.norm(vector_target, np.inf)

3.0

np.linalg.norm(vector_target, 2)

3.7416573867739413

np.linalg.norm(vector_target, 3)

3.3019272488946263

【c】矩阵乘法：@

$\rm [\mathbf{A} *{m\times p}\mathbf{B}* {p\times n}] *{ij} = \sum*{k=1}^p\mathbf{A} *{ik}\mathbf{B}* {kj}$

a = np.arange(4).reshape(-1,2)
a

array([[0, 1],
       [2, 3]])

b = np.arange(-4,0).reshape(-1,2)
b

array([[-4, -3],
       [-2, -1]])

a@b

array([[ -2,  -1],
       [-14,  -9]])