Cholesky decomposition in SciPy - Mini Project: Build & Apply

Cholesky Decomposition with SciPy

📖 Scenario: You work as a data analyst and need to perform a matrix decomposition to simplify solving systems of equations. Cholesky decomposition is a method that breaks down a positive definite matrix into a product of a lower triangular matrix and its transpose. This helps in many data science tasks like optimization and simulations.

🎯 Goal: Learn how to use SciPy to perform Cholesky decomposition on a given positive definite matrix and display the resulting lower triangular matrix.

📋 What You'll Learn

Create a 3x3 positive definite matrix called A with exact values

Create a variable called lower_flag to specify lower triangular output

Use SciPy's cholesky function with A and lower_flag

Print the resulting lower triangular matrix

💡 Why This Matters

🌍 Real World

Cholesky decomposition is used in machine learning for optimization problems, in finance for risk modeling, and in simulations where matrix factorization speeds up calculations.

💼 Career

Data scientists and analysts often use matrix decompositions like Cholesky to simplify complex calculations and improve algorithm performance.

1

Create the matrix A

Create a 3x3 positive definite matrix called A with these exact values: [[4, 12, -16], [12, 37, -43], [-16, -43, 98]]

SciPy

# Create the matrix A with the given values
# Your code here

Need a hint?

This matrix is symmetric and positive definite, perfect for Cholesky decomposition.

2

Set the lower_flag variable

Create a variable called lower_flag and set it to True to get the lower triangular matrix from the decomposition

SciPy

import numpy as np
A = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]])
# Set lower_flag to True
# Your code here

Need a hint?

Setting lower_flag to True tells SciPy to return the lower triangular matrix.

3

Perform Cholesky decomposition

Import cholesky from scipy.linalg and create a variable called L that stores the Cholesky decomposition of A using lower_flag

SciPy

import numpy as np
A = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]])
lower_flag = True
# Import cholesky and compute L
# Your code here

Need a hint?

Use cholesky(A, lower=lower_flag) to get the decomposition.

4

Print the lower triangular matrix

Print the variable L to display the lower triangular matrix from the Cholesky decomposition

SciPy

import numpy as np
A = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]])
lower_flag = True
from scipy.linalg import cholesky
L = cholesky(A, lower=lower_flag)
# Print L
# Your code here

Need a hint?

The printed matrix should be lower triangular with positive diagonal values.