Table of Contents
- Introduction
- What Are Eigenvalues and Eigenvectors?
- The Characteristic Equation
- Computing Eigenvalues and Eigenvectors
- Geometric Interpretation
- Diagonalization and Similarity Transformations
- Defective Matrices and Jordan Form
- Applications of Eigenvalue Problems
- Spectral Theorem
- Hermitian and Unitary Operators
- Eigenvalue Problems in Differential Equations
- Quantum Mechanics and Operators
- Principal Component Analysis (PCA)
- Stability Analysis and Dynamical Systems
- Conclusion
1. Introduction
Eigenvalue problems form the backbone of modern linear algebra and mathematical physics. They reveal hidden structure in linear transformations and are fundamental in solving systems of equations, quantum mechanics, control theory, and data science.
2. What Are Eigenvalues and Eigenvectors?
Given a linear transformation \( A \), an eigenvector \( \vec{v} \neq \vec{0} \) satisfies:
\[
A\vec{v} = \lambda \vec{v}
\]
Where \( \lambda \) is the corresponding eigenvalue.
3. The Characteristic Equation
To find eigenvalues, solve:
\[
\det(A – \lambda I) = 0
\]
This polynomial equation yields the eigenvalues \( \lambda_1, \lambda_2, \dots \).
4. Computing Eigenvalues and Eigenvectors
Steps:
- Solve \( \det(A – \lambda I) = 0 \) for eigenvalues.
- For each \( \lambda \), solve \( (A – \lambda I)\vec{v} = 0 \) to find eigenvectors.
5. Geometric Interpretation
An eigenvector is invariant in direction under the transformation \( A \); it may be stretched or compressed by \( \lambda \), but not rotated.
6. Diagonalization and Similarity Transformations
If \( A \) has \( n \) linearly independent eigenvectors:
\[
A = PDP^{-1}
\]
Where:
- \( D \): diagonal matrix with eigenvalues
- \( P \): matrix of eigenvectors
Diagonalization simplifies powers of \( A \), exponential functions, and more.
7. Defective Matrices and Jordan Form
If \( A \) is not diagonalizable (e.g., not enough linearly independent eigenvectors), it can be put into Jordan canonical form:
\[
A = PJP^{-1}
\]
Where \( J \) contains Jordan blocks for repeated eigenvalues.
8. Applications of Eigenvalue Problems
- Vibrations and modes of mechanical systems
- Stress analysis in engineering
- PageRank algorithm
- Image compression
- Stability analysis in differential equations
9. Spectral Theorem
For real symmetric or complex Hermitian matrices:
- All eigenvalues are real
- There exists an orthonormal basis of eigenvectors
- Matrix is diagonalizable via a unitary (or orthogonal) matrix
10. Hermitian and Unitary Operators
- Hermitian: \( A = A^\dagger \) → real eigenvalues
- Unitary: \( A^\dagger A = I \) → eigenvalues lie on the unit circle
Crucial in quantum mechanics and numerical analysis.
11. Eigenvalue Problems in Differential Equations
Eigenvalues arise when solving linear differential equations:
- Vibrations of strings and membranes
- Heat and wave equations (separation of variables)
- Sturm–Liouville theory
12. Quantum Mechanics and Operators
In quantum theory, observables are represented by operators:
\[
\hat{O} \psi = \lambda \psi
\]
Where \( \psi \) is a wavefunction (eigenvector) and \( \lambda \) is a measurable value.
13. Principal Component Analysis (PCA)
PCA uses eigenvalue decomposition of the covariance matrix:
- Eigenvectors = principal components
- Eigenvalues = variance explained
Used in dimensionality reduction and feature extraction.
14. Stability Analysis and Dynamical Systems
In systems of ODEs:
\[
\dot{x} = Ax
\]
The sign and nature of eigenvalues of \( A \) determine:
- Stability
- Oscillations
- Long-term behavior
15. Conclusion
Eigenvalue problems reveal the intrinsic characteristics of linear systems. Whether describing vibrations, quantum states, or data distributions, the study of eigenvalues and eigenvectors helps us unlock symmetry, stability, and structure in both pure and applied contexts.
Mastering these concepts is essential for modern scientific and engineering disciplines.
