Summary  
This chapter demonstrates how to decompose a matrix into lower and upper triangular factors (LU) and into orthogonal and upper triangular factors (QR) using Python libraries, then verifies each decomposition by reconstructing the original matrix.

General domain of usage  
Numerical linear algebra

Matrix Decomposition Techniques are essential tools in numerical linear algebra, powering solutions for systems of equations, stability analysis, and matrix inversion. 

## Performing LU Decomposition

**LU decomposition** splits a matrix into:

* `L`: lower triangular;
* `U`: upper triangular;
* `P`: permutation matrix to account for row swaps.

import numpy as np
from scipy.linalg import lu

# Define a 2x2 matrix A
A = np.array([[6, 3],
              [4, 3]])

# Perform LU decomposition: P, L, U such that P @ A = L @ U
P, L, U = lu(A)

# Verify that P @ A equals L @ U by reconstructing A from L and U
print(f'L * U:\n{np.dot(L, U)}')

**Why this matters**: LU decomposition is heavily used in numerical methods to solve linear systems and invert matrices efficiently.

## Performing QR Decomposition

QR decomposition factors a matrix into:

* `Q`: Orthogonal matrix (preserves angles/lengths);
* `R`: Upper triangular matrix.

import numpy as np
from scipy.linalg import qr

# Define a 2x2 matrix A
A = np.array([[4, 3],
              [6, 3]])

# Perform QR decomposition: Q (orthogonal), R (upper triangular)
Q, R = qr(A)

# Verify that Q @ R equals A by reconstructing A from Q and R
print(f'Q * R:\n{np.dot(Q, R)}')

**Why this matters**: QR is commonly used for solving least squares problems and is more numerically stable than LU in some scenarios.

What is the role of the permutation matrix `P` in **LU decomposition**?

Suppose you need to solve the system $$A·x = b$$ using **QR decomposition**. What code adjustment would you need to make?

Master the mathematical foundations essential for data science. Explore core concepts in functions, calculus, linear algebra, probability, and dimensionality reduction. Build both theoretical understanding and practical coding experience to strengthen your ability to analyze data, model complex systems, and apply advanced techniques in machine learning.

Explore the foundation of mathematical functions. Learn different types of algebraic and transcendental functions, their properties, and how to implement them in Python to solve real-world problems.

Master the concepts of sets and series, from basic operations to practical applications. Gain hands-on experience implementing set operations and working with arithmetic and geometric series in Python.

Develop a solid understanding of limits, derivatives, integrals, and partial derivatives. Connect theory to practice by implementing these concepts in Python and applying them to optimization through gradient descent.

Build strong knowledge of vectors, matrices, and transformations. Learn decomposition methods and eigenvalue analysis, while reinforcing concepts with Python coding challenges and practical data science applications.

Dive into probability theory and statistics. Study conditional probability, Bayes' theorem, and statistical measures. Implement key concepts in Python, simulate distributions, and strengthen your skills through challenges and quizzes.

Implementing Matrix Decomposition in Python

Performing LU Decomposition

Performing QR Decomposition

1. What is the role of the permutation matrix `P` in LU decomposition?

2. Suppose you need to solve the system $A·x = b$ using QR decomposition. What code adjustment would you need to make?

Implementing Matrix Decomposition in Python

Performing LU Decomposition

Performing QR Decomposition

1. What is the role of the permutation matrix P in LU decomposition?

2. Suppose you need to solve the system A⋅x=bA·x = bA⋅x=b using QR decomposition. What code adjustment would you need to make?

1. What is the role of the permutation matrix `P` in LU decomposition?

2. Suppose you need to solve the system $A·x = b$ using QR decomposition. What code adjustment would you need to make?