NumPy Linear Algebra Cheat Sheet

Reference for NumPy matrix operations, linear algebra functions like inverse, determinant, eigenvalues, and SVD, plus array broadcasting rules.

2 PagesIntermediateMar 10, 2026

Array & Matrix Basics

Create and inspect arrays used as vectors and matrices.

python

import numpy as npA = np.array([[1, 2], [3, 4]])B = np.array([[5, 6], [7, 8]])A.T          # transposeA.shape      # (2, 2)np.zeros((3, 3))np.eye(3)    # identity matrixnp.ones((2, 3))

Core linalg Operations

Matrix multiplication, inverse, determinant, eigenvalues, and solving systems.

python

A = np.array([[4, 2], [1, 3]])np.dot(A, B)              # matrix multiplicationA @ B                     # matrix multiplication (preferred)np.linalg.inv(A)          # matrix inversenp.linalg.det(A)          # determinantnp.linalg.matrix_rank(A)  # rankeigvals, eigvecs = np.linalg.eig(A)   # eigenvalues/eigenvectors# Solve Ax = bb = np.array([1, 2])x = np.linalg.solve(A, b)# Singular Value DecompositionU, S, Vt = np.linalg.svd(A)np.linalg.norm(A)          # Frobenius normnp.linalg.norm(b, ord=2)   # L2 (Euclidean) norm

Broadcasting

Apply operations across arrays of different but compatible shapes.

python

a = np.array([1, 2, 3])          # shape (3,)b = np.array([[1], [2], [3]])    # shape (3, 1)a + b               # broadcasts to shape (3, 3)M = np.random.rand(4, 3)row_means = M.mean(axis=1, keepdims=True)   # shape (4, 1)centered = M - row_means                     # broadcasts across columns

Key Functions

The linalg functions you'll reach for most often.

np.dot / @- Matrix multiplication; use @ for readability with 2D+ arrays
np.linalg.inv- Computes the matrix inverse; raises LinAlgError for singular matrices
np.linalg.solve- Solves Ax = b directly, more stable and faster than computing inv(A) @ b
np.linalg.eig- Returns eigenvalues and eigenvectors of a square matrix
np.linalg.svd- Singular Value Decomposition, used in PCA and dimensionality reduction
broadcasting- NumPy's rule for applying operations across arrays of compatible but different shapes without copying data
np.linalg.norm- Computes vector/matrix norms (L1, L2, Frobenius) for magnitude or regularization calculations

Pro Tip

Prefer np.linalg.solve(A, b) over np.linalg.inv(A) @ b - solving the linear system directly is both faster and numerically more stable than explicitly computing the matrix inverse.

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