Visual Tutorials — Interactive Learning Platform

Dot Product: Matrix × Matrix

Basics

When you have a batch of samples, each sample needs its own layer output. That's a matrix × matrix multiply: each row of X dots with each column of W^T. Here we compute it with a triple loop, then show how np.dot replaces all of it. • A's columns must equal B's rows. • Not commutative: np.dot(A, B) ≠ np.dot(B, A).

X is (3 samples × 3 features), W is (3 neurons × 3 features). X · W^T produces a (3 × 3) output where each row is one sample's neuron outputs, then we add bias b to each row.

Python

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import numpy as np

X = np.array([[ 1.0,  2.0,  3.0],
              [ 2.0,  5.0, -1.0],
              [-1.5,  2.7,  3.3]])
W = np.array([[ 0.2,   0.8,  -0.5 ],
              [ 0.5,  -0.91,  0.26],
              [-0.26, -0.27,  0.17]])
bias = np.array([0.1, -0.2, 0.05])

# ---- Without np.dot (triple loop) ----
result = np.zeros((3, 3))
for i in range(3):          # each sample
    for j in range(3):      # each neuron
        for k in range(3):  # each feature
            result[i,j] += X[i,k] * W[j,k]
        result[i,j] += bias[j]
print("Without np.dot (triple loop):")
print(result)

# ---- With np.dot (one line!) ----
result = np.dot(X, W.T) + bias
print("\nWith np.dot (one line!):")
print(result)

# ---- Not commutative: np.dot(X, W.T) vs np.dot(W, X.T) ----
print("\nCommutativity:")
print("  np.dot(X, W.T) shape:", np.dot(X, W.T).shape, "(samples x neurons)")
print("  np.dot(W, X.T) shape:", np.dot(W, X.T).shape, "(neurons x samples)")
print("  np.dot(X, W.T):\n", np.dot(X, W.T))
print("  np.dot(W, X.T):\n", np.dot(W, X.T))
print("  Different results! Order matters for matrix multiplication.")

# For visualization
a = X
b = W.T

Step 0 / 0

Speed

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Run the operation to see step-by-step explanation.