Linear Algebra Toolkit

Gauss–Jordan Elimination • Eigenvalues • Eigenspaces • Diagonalization

This repository provides a clean, optimized Python toolkit for fundamental linear-algebra operations frequently used in:

• University coursework
• Exam proofs
• Computational mathematics
• Machine learning
• Engineering/scientific computing

It performs:

• ✅ Gauss–Jordan elimination (RREF)
• ✅ Eigenvalues (symbolic)
• ✅ Eigenspaces
• ✅ Diagonalization
• ⚡ Caching optimizations (hashmaps + LRU cache)
• Powered by NumPy + SymPy

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Features

1. Symbolic Eigenvalue Computation

The tool uses SymPy for exact eigenvalues, not floating approximations.

Example:

λ = 4 (multiplicity 2)
λ = 2 (multiplicity 1)

2. Eigenspaces (Basis, Algebraic + Geometric Multiplicities)

For every eigenvalue λ, the script computes:

• Algebraic multiplicity
• Geometric multiplicity
• Basis vectors for the eigenspace

This completely characterizes the linear operator.

3. Automatic Diagonalization

The script attempts to factor the matrix as:

[ A = P D P^{-1} ]

Where:

• D is a diagonal matrix of eigenvalues
• P is the matrix of eigenvectors

If this decomposition fails, the matrix is not diagonalizable.

4. Gauss–Jordan Elimination (RREF)

Useful for:

• Solving linear systems
• Determining independence
• Computing rank
• Proofs involving diagonalizability

5. ⚡ Caching via Hashmaps + functools.lru_cache

To avoid recomputing symbolic eigenvalues (which is expensive), matrices are cached:

@lru_cache(maxsize=32)
def compute_eigen_data(matrix_tuple):
    ...

Matrices are converted to hashable tuples for dictionary-style lookup.

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Installation

git clone https://github.com/<your-username>/codeformaths
cd codeformaths
pip install numpy sympy

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🧪 Usage Example

diagonlize.py

from diagonlize import analyze_matrix

A = [
    [4, 1, 0],
    [0, 4, 0],
    [0, 0, 2]
]

result = analyze_matrix(A)

print(result)

Example Output

--- Eigenvalues ---
λ = 4, multiplicity = 2
λ = 2, multiplicity = 1

--- Eigenspaces ---
Eigenvalue λ = 4
Algebraic multiplicity: 2
Geometric multiplicity: 1
Basis vectors:
     Matrix([[1], [0], [0]])

Eigenvalue λ = 2
Algebraic multiplicity: 1
Geometric multiplicity: 1
Basis vectors:
     Matrix([[0], [0], [1]])

--- Diagonalizable? ---
False

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Mathematical Foundations

1. Eigenvalues

The eigenvalues are roots of the characteristic polynomial:

[ \det(A - \lambda I) = 0 ]

SymPy computes them exactly.

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2. Eigenspaces

Each eigenspace is computed by solving:

[ (A - \lambda I)v = 0 ]

The dimension of this null space is the geometric multiplicity.

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3. Diagonalizability

A matrix is diagonalizable iff:

[ \text{geometric multiplicity of every } \lambda = \text{algebraic multiplicity} ]

Otherwise the matrix fails to have a full set of independent eigenvectors.

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📁 Project Structure

codeformaths/
│
├── diagonlize.py        # Eigenvalues, eigenspaces, diagonalization
├── gauss_jordan.py      # (recommended extension)
├── README.md            # This file
└── examples/            # Optional examples

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License

Open-source under the MIT License.

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Contribute

Feel free to open issues or submit pull requests.

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Acknowledgments

Built using:

• SymPy for symbolic algebra
• NumPy for efficient numerical matrices