Tensor Product¶

import numpy as np

import numpy as np

Vector Product¶

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

print("x * y =", x*y)
print("x @ y =", x@y)
print("sum(x*y)=", np.sum(x*y))

x * y = [ 2  6 12 20]
x @ y = 40
sum(x*y)= 40

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

print("x * y =", x*y)
print("x @ y =", x@y)
print("sum(x*y)=", np.sum(x*y))

x * y = [ 2  6 12 20]
x @ y = 40
sum(x*y)= 40

x = np.random.randint(10, size=(3, 4))
x

array([[5, 2, 3, 2],
       [6, 8, 2, 1],
       [7, 1, 5, 3]])

x = np.random.randint(10, size=(3, 4))
x

array([[5, 2, 3, 2],
       [6, 8, 2, 1],
       [7, 1, 5, 3]])

y = np.random.randint(10, size=(4, 2))
y

array([[8, 7],
       [1, 0],
       [1, 2],
       [5, 6]])

y = np.random.randint(10, size=(4, 2))
y

array([[8, 7],
       [1, 0],
       [1, 2],
       [5, 6]])

x @ y

array([[55, 53],
       [63, 52],
       [77, 77]])

x @ y

array([[55, 53],
       [63, 52],
       [77, 77]])

Let:

Then:

It's defined as:

$$(x\cdot y)(i_1, i_2, j_1, j_2, j_3) = x[i_1, i_2, :, :] \cdot y[:,:,j_1, j_2, j_3]$$

x = np.random.randint(10, size=(3, 2, 1))
x

array([[[6],
        [8]],

       [[2],
        [7]],

       [[7],
        [2]]])

x = np.random.randint(10, size=(3, 2, 1))
x

array([[[6],
        [8]],

       [[2],
        [7]],

       [[7],
        [2]]])

y = np.random.randint(10, size=(2, 1, 2))
y

array([[[2, 3]],

       [[6, 3]]])

y = np.random.randint(10, size=(2, 1, 2))
y

array([[[2, 3]],

       [[6, 3]]])

np.tensordot(x, y).shape

(3, 2)

np.tensordot(x, y).shape

(3, 2)