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Dot Products & Matrix Multiplication (dot, @ operator, matmul)0%
Dot Products & Matrix Multiplication (dot, @ operator, matmul)
Beginner12 min read•Updated: Jul 11, 2026
Vector & Matrix Multiplication: dot(), matmul(), and @
In linear algebra, machine learning (neural networks, linear regression), and 3D computer graphics, matrix multiplication is the foundational building block.
One of the most frequent points of confusion for newcomers to NumPy is distinguishing between element-wise multiplication (*) and formal linear algebraic matrix multiplication (@ or matmul()).
Click to Zoom1. Element-wise vs Matrix Multiplication
- *Element-wise Product (``):** Multiplies elements at matching positions. Arrays must have compatible shapes for broadcasting.
- Matrix Multiplication (
@ornp.matmul): Computes row-by-column inner dot products according to formal linear algebra rules:
$$C_{ij} = \sum_{k=1}^K A_{ik} B_{kj}$$
For matrices $A$ of shape $(M, K)$ and $B$ of shape $(K, N)$, the inner dimensions ($K$) must match exactly, and the resulting matrix $C$ has shape $(M, N)$.
import numpy as np
A = np.array([[1, 2],
[3, 4]])
B = np.array([[5, 6],
[7, 8]])
# 1. Element-wise (Hadamard) product:
print("Element-wise (A * B):
", A * B)
# [[1*5, 2*6], [3*7, 4*8]] = [[5, 12], [21, 32]]
# 2. Formal Matrix Multiplication using @ operator (Python 3.5+ standard):
print("
Matrix Multiplication (A @ B):
", A @ B)
# [[1*5 + 2*7, 1*6 + 2*8], [3*5 + 4*7, 3*6 + 4*8]] = [[19, 22], [43, 50]]
2. The Difference Between np.dot() and np.matmul()
NumPy provides both np.dot() and np.matmul() (or the @ operator). For 2D matrices, they produce identical results. However, their behaviors diverge on 1D vectors and higher-order tensors:
A. 1D Vector Dot Product (Inner Product)
For 1D vectors, both compute the standard scalar inner product:
v1 = np.array([1, 2, 3])
v2 = np.array([4, 5, 6])
# 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32
print("Vector inner product:", np.dot(v1, v2)) # 32
print("Using @ operator: ", v1 @ v2) # 32
B. Higher-Order Tensors and Batch Matrix Multiplication
This is where matmul() / @ shines and is the standard for deep learning:
np.matmul()treats trailing 2 dimensions as matrices and broadcasts over leading batch dimensions.np.dot()computes a sum-product over the last axis of $A$ and second-to-last axis of $B$, which is not batch matrix multiplication!
# Batch of 10 matrices, each 3x4
batch_A = np.random.randn(10, 3, 4)
# Batch of 10 matrices, each 4x5
batch_B = np.random.randn(10, 4, 5)
# Matmul performs 10 simultaneous 3x5 matrix multiplications:
batch_result = batch_A @ batch_B
print("Batch matmul shape:", batch_result.shape) # (10, 3, 5)
3. Vector-Matrix Multiplication
When multiplying a 1D vector $v$ with a 2D matrix $M$:
- If $v$ is on the left (
v @ M), NumPy temporarily promotes $v$ to a row matrix $(1, K)$, performs multiplication, and squeezes the output back to a 1D vector. - If $v$ is on the right (
M @ v), NumPy promotes $v$ to a column matrix $(K, 1)$, performs multiplication, and squeezes back:
M = np.array([[1, 2, 3],
[4, 5, 6]]) # shape (2, 3)
v = np.array([10, 20, 30]) # shape (3,)
result = M @ v
print("M @ v shape:", result.shape) # (2,)
print("M @ v result:", result) # [1*10 + 2*20 + 3*30, 4*10 + 5*20 + 6*30] = [140, 320]
Interactive Quiz
Multiple Choice Questions
Question 1
Which Python operator is designated for matrix multiplication?
Question 2
If matrix $A$ has shape $(4, 7)$ and matrix $B$ has shape $(7, 3)$, what is the shape of $A @ B$?
Question 3
What is the fundamental difference between
np.matmul and np.dot on 3D tensors?Question 4
Given
a = np.array([1, 2]) and b = np.array([3, 4]), what is a * b?Question 5
Why does multiplying a matrix of shape (3, 5) with a matrix of shape (3, 5) using
A @ B raise a ValueError?Next Lesson
Matrix Determinant, Inversion & Solving Linear Systems
Continue learning with hands-on practice, examples, and exercises in the upcoming topic.
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Practice Quiz
Test your understanding of this lesson with 5 questions. Each question has one correct answer.