¶ Eigenvalues and Eigenvectors
¶ 100 Chapter Titles on Eigenvalues and Eigenvectors (Beginner to Advanced)
Here are 100 chapter titles on eigenvalues and eigenvectors, progressing from beginner to advanced levels, covering various aspects of the topic:
I. Foundations & Basic Concepts (1-20)
- Introduction to Vectors and Matrices
- Matrix Operations: Addition, Multiplication, and Transpose
- Linear Transformations: A Visual Introduction
- What are Eigenvalues and Eigenvectors?
- The Eigenvalue Equation: Ax = λx
- Geometric Interpretation of Eigenvectors
- Eigenvalues and Eigenvectors of 2x2 Matrices
- Calculating Eigenvalues: The Characteristic Polynomial
- Finding Eigenvectors: Solving the System of Equations
- Examples: Eigenvalues and Eigenvectors in Action
- Properties of Eigenvalues and Eigenvectors
- Uniqueness of Eigenvectors (up to a scalar multiple)
- Eigenvalues of Triangular Matrices
- Eigenvalues of Diagonal Matrices
- Eigenvalues and the Trace of a Matrix
- Eigenvalues and the Determinant of a Matrix
- Complex Eigenvalues and Eigenvectors
- Eigenvalues of Real Symmetric Matrices (Introduction)
- Applications: A First Look
- Practice Problems: Basic Eigenvalue and Eigenvector Calculations
II. Advanced Concepts & Techniques (21-40)
- The Characteristic Polynomial: A Deeper Dive
- Cayley-Hamilton Theorem: Statement and Implications
- Minimal Polynomial and its Relation to Eigenvalues
- Generalized Eigenvectors and Eigenspaces
- Similarity Transformations and Invariant Subspaces
- Diagonalization of Matrices: Conditions and Procedures
- Matrix Diagonalization: Examples and Applications
- Jordan Canonical Form: Introduction
- Computing Jordan Forms: A Step-by-Step Guide
- Functions of Matrices: Using Eigenvalue Decomposition
- The Spectral Theorem for Normal Matrices
- Unitary Matrices and their Eigenvalues
- Hermitian Matrices and their Properties
- Positive Definite Matrices and their Eigenvalues
- The Courant-Fischer Minimax Theorem
- Gershgorin Circle Theorem: Locating Eigenvalues
- Perturbation Theory for Eigenvalues
- Sensitivity of Eigenvalues to Matrix Entries
- Applications: Differential Equations (First Order)
- Practice Problems: Advanced Eigenvalue and Eigenvector Calculations
III. Applications in Various Fields (41-60)
- Eigenvalues and Eigenvectors in Linear Systems of Differential Equations
- Stability Analysis of Linear Systems
- Eigenvalues and Eigenvectors in Vibrational Analysis
- Normal Modes and Natural Frequencies
- Eigenvalues and Eigenvectors in Quantum Mechanics
- The Schrödinger Equation and Eigenstates
- Eigenvalues and Eigenvectors in Data Analysis: PCA
- Principal Component Analysis: A Detailed Look
- Eigenvalues and Eigenvectors in Image Processing
- Applications: PageRank Algorithm
- Eigenvalues and Eigenvectors in Markov Chains
- Steady-State Vectors and Limiting Probabilities
- Eigenvalues and Eigenvectors in Graph Theory
- Spectral Graph Theory: Basic Concepts
- The Laplacian Matrix and its Eigenvalues
- Eigenvalues and Eigenvectors in Economics
- Applications: Leslie Matrix and Population Growth
- Eigenvalues and Eigenvectors in Finance
- Portfolio Optimization using Eigenvalues
- Eigenvalues and Eigenvectors in Machine Learning
IV. Numerical Methods (61-80)
- Numerical Computation of Eigenvalues: Introduction
- The Power Iteration Method: Finding the Dominant Eigenvalue
- Inverse Iteration: Finding the Smallest Eigenvalue
- The QR Algorithm: Basic Version
- The QR Algorithm with Shifts
- The Jacobi Method for Symmetric Matrices
- Tridiagonalization: A Preprocessing Step
- The Bisection Method for Eigenvalues of Symmetric Tridiagonal Matrices
- Divide and Conquer Algorithm for Eigenvalues
- Lanczos Algorithm: For Large Sparse Matrices
- Arnoldi Iteration: For Large Non-Symmetric Matrices
- Krylov Subspace Methods for Eigenvalue Problems
- Convergence Analysis of Numerical Eigenvalue Algorithms
- Error Estimation and Refinement Techniques
- Software Libraries for Eigenvalue Computations (e.g., LAPACK)
- Parallel Computation of Eigenvalues
- Dealing with Ill-Conditioned Eigenvalue Problems
- Eigenvalue Computation for Structured Matrices
- Applications: Numerical Solution of Differential Equations
- Practice Problems: Numerical Eigenvalue Computations
V. Advanced Topics and Research Directions (81-100)
- Generalized Eigenvalue Problem: Ax = λBx
- The QZ Algorithm for Generalized Eigenvalue Problems
- Non-Normal Matrices and their Eigenvalues
- Pseudo-Eigenvalues and Pseudospectra
- Structured Eigenvalue Problems: Exploiting Matrix Structure
- Parameter-Dependent Eigenvalue Problems
- Bifurcation Analysis of Eigenvalues
- Eigenvalues and Eigenvectors of Infinite-Dimensional Operators
- Spectral Theory of Linear Operators
- Applications: Functional Analysis and Operator Theory
- Eigenvalues and Eigenvectors in Control Theory
- Linear Systems and their Transfer Functions
- Eigenvalues and Eigenvectors in Optimization
- Convex Optimization and Eigenvalue Problems
- Eigenvalues and Eigenvectors in Signal Processing
- Time-Frequency Analysis and Eigenfunctions
- Research Topics: Recent Advances in Eigenvalue Computations
- Open Problems in Eigenvalue Theory
- Connections to Other Areas of Mathematics
- The Future of Eigenvalue Research