What is 3D Matrix Multiplication?
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.
Here we will see two different examples of matrix multiplication where we have used different dimensions in each example.
Multiplication of matrix A of shape (3,3,2) with 3D matrix B of shape (3,2,4)
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).
Python3
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)) C = np.matmul(A, B) print ( "Product C:\n{}, shape={}" . format (C, C.shape)) |
Output:
Multiplication of matrix A of shape (3,5,2) with 3D matrix B of shape (3,2,5)
The code generates two random integer matrices A and B, then multiplies them using matrix multiplication. It prints the shapes and content of these matrices. Matrix multiplication is a fundamental operation in linear algebra and the code demonstrates how to perform it using NumPy in Python.
Python3
import numpy as np np.random.seed( 42 ) A = np.random.randint( 0 , 10 , size = ( 3 , 5 , 2 )) B = np.random.randint( 0 , 10 , size = ( 3 , 2 , 5 )) print ( "A:\n{}, shape={}\nB:\n{}, shape={}" . format ( A, A.shape, B, B.shape)) C = np.matmul(A, B) print ( "Product C:\n{}, shape={}" . format (C, C.shape)) |
Output:
NumPy – 3D matrix multiplication
Numpy is a general-purpose array-processing package. It provides a high-performance multidimensional array object and tools for working with these arrays. It is the fundamental package for scientific computing with Python.