Matrix Product

Matrix multiplication (the matrix product) is a fundamental operation in linear algebra that combines two matrices to produce a new one - and it powers core calculations behind Linear Regression, Neural Networks, and many other Machine Learning algorithms.

Matrix Product vs Element-Wise Multiplication

It's important not to confuse the matrix product with simple element-wise multiplication (using *), which just multiplies corresponding elements. The matrix product follows specific linear algebra rules instead.

import numpy as np

a = np.array([[1, 2], [3, 4]])
b = np.array([[5, 6], [7, 8]])

print(a * b)     # element-wise: [[5, 12], [21, 32]]
print(a @ b)     # matrix product: [[19, 22], [43, 50]]
print(np.dot(a, b))   # same result as a @ b

How Matrix Multiplication Works

For two matrices to be multiplied, the number of columns in the first matrix must equal the number of rows in the second. Each element in the result is calculated as the sum of products between a row from the first matrix and a column from the second.

import numpy as np

a = np.array([[1, 2, 3]])        # shape (1, 3)
b = np.array([[4], [5], [6]])    # shape (3, 1)

result = a @ b
print(result)   # [[32]]  ->  (1*4 + 2*5 + 3*6)

Coming Up Next

Next, you'll explore more useful NumPy functions for sorting, searching, and summarizing array data.