einsum愛因斯坦求和約定

愛因斯坦求和約定（Einstein Notation）
在數(shù)學(xué)中涎劈，愛因斯坦求和約定是一種標(biāo)記法算撮，也稱為Einstein Summation Convention轩触，在處理關(guān)于坐標(biāo)的方程式時(shí)十分有效壹蔓。簡單來說杭攻，愛因斯坦求和就是簡化掉求和式中的求和符號(hào) ,這樣就會(huì)使公式更加簡潔

轉(zhuǎn)置
${B_{ji}} = {A_{ij}}$

import numpy as np
a = np.arange(0, 9).reshape(3, 3)
print(a)
b = np.einsum('ij->ji', a)
print(b)

Output:
a: [[0 1 2]
 [3 4 5]
 [6 7 8]]
b: [[0 3 6]
 [1 4 7]
 [2 5 8]]

全部元素求和
$sum{\rm{ = }}\sum\limits_i {\sum\limits_j {{A_{ij}}} }$

import numpy as np
a = np.arange(0, 9).reshape(3, 3)
print(a)
b = np.einsum('ij->', a)
print(b)

Output:
a: [[0 1 2]
 [3 4 5]
 [6 7 8]]
b: 36

某一維度求和
$sum{\rm{ = }}\sum\limits_i {{A_{ij}}}$

import numpy as np
a = np.arange(0, 9).reshape(3, 3)
print(a)
b = np.einsum('ij->i', a)
print(b)

Output:
a: [[0 1 2]
 [3 4 5]
 [6 7 8]]
b: [ 3 12 21]

矩陣對應(yīng)維度相乘(廣播形式)
${C_{ij}} = {A_{ij}}{B_j}$

import numpy as np
a = np.arange(0, 12).reshape(3, 4)
print(a)
b = np.arange(0, 4).reshape(4)
print(b)
c = np.einsum('ij,j->ij', a, b)
print(c)

Output:
a: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
b: [0 1 2 3]
c: [[ 0  1  4  9]
 [ 0  5 12 21]
 [ 0  9 20 33]]

矩陣對應(yīng)維度相乘(求和形式)
${C_i} = \sum\limits_j {{A_{ij}}{B_j}}$

import numpy as np
a = np.arange(0, 12).reshape(3, 4)
print(a)
b = np.arange(0, 4).reshape(4)
print(b)
c = np.einsum('ij,j->i', a, b)
print(c)

Output:
a: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
b: [0 1 2 3]
c: [14 38 62]

矩陣點(diǎn)積
$C = \sum\limits_i {\sum\limits_j {{A_{ij}}{B_{ij}}} }$

import numpy as np
a = np.arange(0, 12).reshape(3, 4)
print(a)
b = np.arange(0, 12).reshape(3, 4)
print(b)
c = np.einsum('ij,ij->', a, b)
print(c)

Output:
a: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
b: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
c: 506

矩陣外積(相乘)
${C_{ij}} = \sum\limits_k {{A_{ik}}{B_{kj}}}$

import numpy as np
a = np.arange(0, 12).reshape(3, 4)
print(a)
b = np.arange(0, 12).reshape(4, 3)
print(b)
c = np.einsum('ik,kj->ij', a, b)
print(c)

Output:
a: [[ 0  1  2  3]
 [ 4  5  6  7]
 [ 8  9 10 11]]
b: [[ 0  1  2]
 [ 3  4  5]
 [ 6  7  8]
 [ 9 10 11]]
c: [[ 42  48  54]
 [114 136 158]
 [186 224 262]]

https://zhuanlan.zhihu.com/p/44954540

最后編輯于：2021.05.28 10:21:32

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