Matrix Decomposition Algorithms 💬

Welcome to the nlp-2.1-matrix-decomposition repository! This project provides a collection of algorithms for matrix decomposition, a fundamental concept in linear algebra. Whether you're working on data analysis, machine learning, or scientific computing, understanding these algorithms can enhance your skills and broaden your toolkit.

Table of Contents

1. Introduction
2. Matrix Decomposition Overview
3. Algorithms Included
   • Eigen Decomposition
   • LU Decomposition
   • PLU Decomposition
   • QR Decomposition
   • Singular Value Decomposition (SVD)
   • Spectral Decomposition
4. Installation
5. Usage
6. Examples
7. Contributing
8. License
9. Contact

Introduction

Matrix decomposition plays a vital role in various fields such as statistics, computer science, and engineering. This repository aims to provide a straightforward implementation of key matrix decomposition algorithms. By using these algorithms, you can simplify complex problems and gain insights into data structures.

Matrix Decomposition Overview

Matrix decomposition involves breaking down a matrix into simpler, constituent matrices. This process can help solve systems of equations, perform dimensionality reduction, and enhance data visualization. The main types of matrix decomposition include:

• Eigen Decomposition
• LU Decomposition
• PLU Decomposition
• QR Decomposition
• Singular Value Decomposition (SVD)
• Spectral Decomposition

Each of these techniques has its own applications and benefits, making them essential for anyone working with matrices.

Algorithms Included

Eigen Decomposition

Eigen decomposition decomposes a square matrix into its eigenvalues and eigenvectors. This technique is crucial for understanding the properties of matrices, especially in the context of transformations and stability analysis.

Key Concepts:

• Eigenvalues: Scalars that indicate how much the eigenvectors are stretched or compressed during a transformation.
• Eigenvectors: Directions in which the transformation acts.

LU Decomposition

LU decomposition factors a matrix into a lower triangular matrix (L) and an upper triangular matrix (U). This method is particularly useful for solving linear equations.

Key Concepts:

• Lower Triangular Matrix (L): Contains all zeros above the main diagonal.
• Upper Triangular Matrix (U): Contains all zeros below the main diagonal.

PLU Decomposition

PLU decomposition extends LU decomposition by adding a permutation matrix (P). This method improves numerical stability and allows for the solution of more complex systems.

Key Concepts:

• Permutation Matrix (P): A matrix that rearranges the rows of another matrix.

QR Decomposition

QR decomposition breaks a matrix into an orthogonal matrix (Q) and an upper triangular matrix (R). This method is often used in solving linear least squares problems.

Key Concepts:

• Orthogonal Matrix (Q): A matrix whose columns are orthogonal unit vectors.
• Upper Triangular Matrix (R): Similar to LU decomposition.

Singular Value Decomposition (SVD)

SVD is a powerful technique that decomposes a matrix into three other matrices, revealing its structure. It is widely used in statistics, signal processing, and machine learning.

Key Concepts:

• Singular Values: Non-negative values that provide insight into the matrix's properties.
• U and V Matrices: Orthogonal matrices that correspond to the left and right singular vectors.

Spectral Decomposition

Spectral decomposition expresses a matrix in terms of its eigenvalues and eigenvectors. This method is particularly useful for symmetric matrices.

Key Concepts:

• Symmetric Matrix: A matrix that is equal to its transpose.

Installation

To get started with this repository, follow these simple steps:

1. Clone the repository:

   git clone https://github.com/reianrafi/nlp-2.1-matrix-decomposition.git
   

2. Navigate to the project directory:

   cd nlp-2.1-matrix-decomposition
   

3. Install the required dependencies. If you're using Python, you can do this with pip:

   pip install -r requirements.txt
   

Usage

Once you have installed the repository, you can start using the algorithms. Each algorithm has its own module, and you can import them as needed. For example:

from lu_decomposition import LU

Make sure to check the documentation for each algorithm to understand its parameters and return values.

Examples

Here are some examples to illustrate how to use the algorithms:

Eigen Decomposition Example

import numpy as np
from eigen_decomposition import EigenDecomposition

matrix = np.array([[4, -2], [1, 1]])
eigen = EigenDecomposition(matrix)
values, vectors = eigen.compute()
print("Eigenvalues:", values)
print("Eigenvectors:", vectors)

LU Decomposition Example

import numpy as np
from lu_decomposition import LU

matrix = np.array([[3, 2, 1], [6, 5, 4], [1, 0, 3]])
lu = LU(matrix)
L, U = lu.decompose()
print("Lower Triangular Matrix (L):", L)
print("Upper Triangular Matrix (U):", U)

SVD Example

import numpy as np
from svd import SVD

matrix = np.array([[1, 2], [3, 4]])
svd = SVD(matrix)
U, S, V = svd.decompose()
print("U Matrix:", U)
print("Singular Values:", S)
print("V Matrix:", V)

Contributing

We welcome contributions to improve this repository. If you have suggestions, bug fixes, or new features, please follow these steps:

1. Fork the repository.
2. Create a new branch for your feature or bug fix.
3. Make your changes and commit them.
4. Push to your branch.
5. Create a pull request.

Please ensure your code adheres to the project's coding standards.

License

This project is licensed under the MIT License. See the LICENSE file for details.

Contact

For questions or feedback, feel free to reach out:

• Email: your-email@example.com
• GitHub: reianrafi

Thank you for visiting the nlp-2.1-matrix-decomposition repository! For the latest updates, check the Releases section.