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Cholesky Decomposition Solver — Course & Implementation

🚀 Professional course project for understanding and implementing Cholesky Decomposition from scratch in Python.

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🔥 Project Overview

This repository provides a complete implementation and mathematical explanation of Cholesky Decomposition for symmetric positive definite matrices.

It includes:

• Mathematical derivation
• Algorithm explanation
• Python implementation from scratch
• Numerical stability discussion
• Practical applications in linear systems

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Keywords

cholesky decomposition
cholesky factorization
cholesky solver python
matrix decomposition
numerical linear algebra
positive definite matrix
linear system solver
linalg cholesky from scratch
least squares optimization
python cholesky implementation

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📚 Mathematical Definition

For a symmetric positive definite matrix:

$$ A \in \mathbb{R}^{n \times n} $$

Cholesky decomposition states:

$$ A = LL^T $$

Where:

• $$L$$ — lower triangular matrix
• $$L^T$$ — transpose of $$L$$

Condition:

$$ A = A^T $$

and

$$ x^T A x > 0 \quad \forall x \neq 0 $$

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🧠 Elementwise Formula

The elements of matrix $$L$$ are computed as:

For diagonal:

$$ L_{ii} = \sqrt{A_{ii} - \sum_{k=1}^{i-1} L_{ik}^2} $$

For off-diagonal:

$$ L_{ij} = \frac{1}{L_{jj}} \left( A_{ij} - \sum_{k=1}^{j-1} L_{ik} L_{jk} \right) $$

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⚡ Applications

Cholesky decomposition is used in:

• Solving linear systems
• Gaussian processes
• Bayesian statistics
• Optimization
• Covariance matrix factorization
• Machine learning models

It is computationally cheaper than LU for SPD matrices.

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🏗 Project Structure

cholesky-decomposition-solver/
│
├── README.md
├── LICENSE
├── requirements.txt
│
├── src/
│   ├── cholesky.py
│   ├── solver.py
│
├── examples/
│   └── demo.py
│
├── docs/
│   └── theory.md
│
├── images/
│   └── factorization.png
│
└── index.html

Clean structure improves:

✔ Professional appearance
✔ Search visibility
✔ Educational clarity

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🐍 Example Implementation

Basic Cholesky Algorithm

import numpy as np

def cholesky_decomposition(A):
    A = A.astype(float)
    n = A.shape[0]
    L = np.zeros_like(A)

    for i in range(n):
        for j in range(i + 1):
            s = sum(L[i, k] * L[j, k] for k in range(j))

            if i == j:
                L[i, j] = np.sqrt(A[i, i] - s)
            else:
                L[i, j] = (A[i, j] - s) / L[j, j]

    return L

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🚀 Installation

pip install -r requirements.txt

Run example:

python examples/demo.py