The Eigenvalue Odyssey

Table of Contents

1. Unveiling the Basics
- Defining Eigenvalues and Eigenvectors
- Understanding Linear Transformations
- Introduction to Matrices and Determinants

2. Mathematical Preliminaries
- Complex Numbers and Their Properties
- Polynomials and their Roots
- In-Depth Look at Determinants

3. Computational Techniques
- Characteristic Polynomial Approach
- Power Iteration Method
- QR Algorithm and Its Applications

4. Special Matrices
- Diagonal and Triangular Matrices
- Symmetric and Hermitian Matrices
- Sparse and Dense Matrix Considerations

5. Spectral Theorems
- The Spectrum of a Matrix
- Rayleigh Quotient and Min-Max Theorem
- Functional Calculus for Operators

6. Practical Applications
- Engineering and The Eigenvalue Problem
- Eigenvalues in Physics: Quantum Mechanics
- Eigenvectors in Computer Graphics

7. Advanced Computational Methods
- Jacobi and Givens Rotation Methods
- Lanczos Algorithm for Large Matrices
- Divide and Conquer Techniques

8. Perturbation Theory
- Fundamentals of Perturbation
- Eigenvalue Sensitivity Analysis
- Applications in Stability and Chaos

9. Numerical Stability and Errors
- Round-Off Errors and Ill-Conditioned Matrices
- Error Analysis in Eigenvalue Computation
- Improving Numerical Stability

10. Software and Programming
- Using MATLAB for Eigenvalue Problems
- Python Libraries: NumPy and SciPy
- Custom Algorithms in C/C++

11. Case Studies in Eigenvalue Calculations
- Vibration Analysis and Mechanical Systems
- Stability of Structures and Eigenvalues
- Graph Theory and Network Analysis

12. Expanding Horizons
- Generalized Eigenvalue Problems
- Tensor Decompositions and their Eigenvalues
- Quantum Computing and Eigen Decomposition