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?

This course distills the essentials of Python's math module, focusing on trigonometric functions, logarithmic operations, and mathematical constants. Learners will gain hands-on experience using Python to perform these standard mathematical computations, ideal for beginners seeking practical skills with the math module.

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?