How to Use np.linalg in NumPy for Linear Algebra

Use np.linalg in NumPy to perform linear algebra tasks such as matrix inversion, solving linear systems, and finding eigenvalues. Import NumPy and call functions like np.linalg.inv() for inverse or np.linalg.solve() to solve equations.

Syntax

The np.linalg module contains functions for linear algebra operations. Common functions include:

• np.linalg.inv(a): Computes the inverse of a square matrix a.
• np.linalg.solve(a, b): Solves the linear system ax = b for x.
• np.linalg.det(a): Calculates the determinant of matrix a.
• np.linalg.eig(a): Returns eigenvalues and eigenvectors of a.

Each function takes arrays as input and returns arrays or scalars as output.

python

import numpy as np

# Inverse of matrix a
inv_a = np.linalg.inv(a)

# Solve linear system ax = b
x = np.linalg.solve(a, b)

# Determinant of matrix a
det_a = np.linalg.det(a)

# Eigenvalues and eigenvectors of a
eigenvalues, eigenvectors = np.linalg.eig(a)

Example

This example shows how to invert a matrix, solve a system of equations, and compute the determinant using np.linalg.

python

import numpy as np

# Define a 2x2 matrix
A = np.array([[3, 1], [1, 2]])

# Define vector b
b = np.array([9, 8])

# Calculate inverse of A
A_inv = np.linalg.inv(A)

# Solve Ax = b
x = np.linalg.solve(A, b)

# Calculate determinant of A
det_A = np.linalg.det(A)

print("Inverse of A:\n", A_inv)
print("Solution x of Ax = b:\n", x)
print("Determinant of A:", det_A)

Output

Inverse of A:
 [[ 0.4 -0.2]
 [-0.2  0.6]]
Solution x of Ax = b:
 [2. 3.]
Determinant of A: 5.0

Common Pitfalls

Common mistakes when using np.linalg include:

• Trying to invert a non-square or singular matrix, which causes errors.
• Passing arrays with wrong shapes to np.linalg.solve.
• Ignoring that some functions expect float or complex types for accurate results.

Always check matrix shape and properties before applying linear algebra functions.

python

import numpy as np

# Wrong: Non-square matrix inversion
try:
    A = np.array([[1, 2, 3], [4, 5, 6]])
    np.linalg.inv(A)
except np.linalg.LinAlgError as e:
    print("Error:", e)

# Right: Square matrix inversion
A = np.array([[1, 2], [3, 4]])
inv_A = np.linalg.inv(A)
print("Inverse of A:\n", inv_A)

Output

Error: Last 2 dimensions of the array must be square
Inverse of A:
 [[-2.   1. ]
 [ 1.5 -0.5]]

Quick Reference

Here is a quick reference for common np.linalg functions:

Function	Description	Input	Output
np.linalg.inv(a)	Inverse of square matrix a	2D square array	2D array (inverse)
np.linalg.solve(a, b)	Solve linear system ax = b	2D square array a, 1D or 2D array b	Array x solution
np.linalg.det(a)	Determinant of matrix a	2D square array	Scalar determinant
np.linalg.eig(a)	Eigenvalues and eigenvectors	2D square array	Tuple (eigenvalues, eigenvectors)
np.linalg.norm(x)	Vector or matrix norm	Array	Scalar norm

Key Takeaways

Use np.linalg functions for common linear algebra tasks like inversion, solving, and eigen computations.

Ensure matrices are square and non-singular before inverting or solving systems.

np.linalg.solve is preferred over manual inversion for solving linear equations.

Check input shapes carefully to avoid errors.

np.linalg provides fast, reliable linear algebra tools integrated with NumPy arrays.