A 3D matrix is nothing but a collection (or a stack) of many 2D matrices, just like how a 2D matrix is a collection/stack of many 1D vectors. So, matrix multiplication of 3D matrices involves multiple multiplications of 2D matrices, which eventually boils down to a dot product between their row/column vectors.
Let us consider an example matrix A of shape (3,3,2) multiplied with another 3D matrix B of shape (3,2,4).
Python
import numpy as np np.random.seed(42) A = np.random.randint(0, 10, size=(3, 3, 2))B = np.random.randint(0, 10, size=(3, 2, 4)) print("A:\n{}, shape={}\nB:\n{}, shape={}".format( A, A.shape, B, B.shape)) |
OUTPUT:
The first matrix is a stack of three 2D matrices each of shape (3,2), and the second matrix is a stack of 3 2D matrices, each of shape (2,4).
The matrix multiplication between these two will involve three multiplications between corresponding 2D matrices of A and B having shapes (3,2) and (2,4) respectively. Specifically, the first multiplication will be between A[0] and B[0], the second multiplication will be between A[1] and B[1], and finally, the third multiplication will be between A[2] and B[2]. The result of each individual multiplication of 2D matrices will be of shape (3,4). Hence, the final product of the two 3D matrices will be a matrix of shape (3,3,4).
Let’s realize this using code.
Python
C = np.matmul(A, B) print("Product C:\n{}, shape={}".format(C, C.shape)) |
Output:
