LU Decomposition

What is LU Decomposition?

LU Decomposition means breaking a matrix A into two matrices: L (Lower Triangular) and U (Upper Triangular), such that:

A = L × U

This is useful because working with triangular matrices makes solving equations easier and faster.

How Does It Work?

Let’s say we want to solve the system Ax = b. Using LU decomposition:

1. First, decompose A into L and U: A = L × U
2. Rewrite the equation: LUx = b
3. Let Ux = y and solve Ly = b using forward substitution
4. Then solve Ux = y using backward substitution

🔍 Example:

Given:
A = [[2, 3], [4, 7]]
We can write:
L = [[1, 0], [2, 1]], U = [[2, 3], [0, 1]]

✅ So: A = L × U

Applications of LU Decomposition

• 📐 Efficiently solve systems of linear equations
• 📊 Used in numerical analysis and engineering simulations
• 📈 Used in computer programs for matrix inversion and determinants
• 🧠 Basis for many advanced techniques in machine learning and optimization

🔗 LU decomposition is a powerful tool when working with large systems where direct methods like Gaussian elimination are inefficient.