Matrix Multiplication in Numpy

import numpy as np

A = np.random.rand(3, 4) # 3 x 4 matrix
print(A)

[[0.91093404 0.9512029  0.37012699 0.49695407]
 [0.12733269 0.98029049 0.3460466  0.83258018]
 [0.44438983 0.89992234 0.74756983 0.10297354]]

B = np.random.rand(4, 2)
print(B)

[[0.56647329 0.45461109]
 [0.25323378 0.15791856]
 [0.54631074 0.61116914]
 [0.55152014 0.21768352]]

Define matrix multiplication of an $M \times K$ matrix $A$ by a $K \times N$ matrix $B$ as the product $AB$, which is an $M \times N$ matrix, where

$AB[i, j] = A[i, :] \cdot B[:, j]$

The element $AB_{ij}$ is the dot product of the $i^{th}$ row of $A$ and the $j^{th}$ column of $B$

Below are a couple of examples

Example1

A $5 \times 3$ matrix multiplied by a $3 \times 4$ matrix yields a $5 \times 4$ matrix. For example, row 3, column 4 can be obtained as follows:

Here's an animation showing how to obtain the rest of the rows and columns (Click here to watch a youtube video where you can pause the frames if needed)

Example2

A $2 \times 4$ matrix multiplied by a $4 \times 3$ matrix yields a $2 \times 3$ matrix (Click here to watch a version of this video on youtube where you can pause the frames if needed)

def matrixmul(A, B):
    M = A.shape[0]
    K = A.shape[1]
    assert(B.shape[0] == K)
    N = B.shape[1]
    AB = np.zeros((M, N))
    for i in range(M):
        for j in range(N):
            AB[i, j] = np.sum(A[i, :]*B[:, j])
    return AB

AB = matrixmul(A, B)
print(AB)

[[1.23318104 0.89872221]
 [0.96860695 0.6054249 ]
 [0.94482312 0.82344624]]

AB2 = np.dot(A, B)
print(AB2)

[[1.23318104 0.89872221]
 [0.96860695 0.6054249 ]
 [0.94482312 0.82344624]]