Applications of Eigenvalues

Overview

Eigenvalues and eigenvectors have countless applications across mathematics, science, and engineering. They simplify complex problems by revealing the fundamental modes of a system.

Matrix Powers

If $A$ is diagonalizable with $A = PDP^{-1}$:

$A^n = PD^n P^{-1}$

where:

$D^n = \begin{bmatrix} \lambda_1^n & 0 & \cdots & 0 \\ 0 & \lambda_2^n & \cdots & 0 \\ \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & \cdots & \lambda_n^n \end{bmatrix}$

Example

$A = \begin{bmatrix} 4 & 2 \\ 1 & 3 \end{bmatrix}, \quad P = \begin{bmatrix} 2 & 1 \\ 1 & -1 \end{bmatrix}, \quad D = \begin{bmatrix} 5 & 0 \\ 0 & 2 \end{bmatrix}$

$A^3 = PD^3 P^{-1} \text{ where } D^3 = \begin{bmatrix} 125 & 0 \\ 0 & 8 \end{bmatrix}$

Fibonacci Numbers

The Fibonacci sequence $F_{n+2} = F_{n+1} + F_n$ can be expressed as:

$\begin{bmatrix} F_{n+1} \\ F_n \end{bmatrix} = \begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}^n \begin{bmatrix} 1 \\ 0 \end{bmatrix}$

Eigenvalues of $\begin{bmatrix} 1 & 1 \\ 1 & 0 \end{bmatrix}$:

$\lambda_1 = \frac{1+\sqrt{5}}{2} \approx 1.618 \text{ (golden ratio } \varphi)$

$\lambda_2 = \frac{1-\sqrt{5}}{2} \approx -0.618$

Closed form (Binet's formula):

$F_n = \frac{\varphi^n - \psi^n}{\sqrt{5}}$

Markov Chains

For transition matrix $P$:

• $\lambda = 1$ is always an eigenvalue
• Stationary distribution is eigenvector for $\lambda = 1$
• Long-term behavior determined by eigenvalue 1

Example

$P = \begin{bmatrix} 0.8 & 0.3 \\ 0.2 & 0.7 \end{bmatrix} \quad \text{(transition probabilities)}$

Stationary: solve $(P - I)\boldsymbol{\pi} = \mathbf{0}$

$\boldsymbol{\pi} = \begin{bmatrix} 0.6 \\ 0.4 \end{bmatrix}$ (60% state 1, 40% state 2)

Differential Equations

For $\frac{d\mathbf{x}}{dt} = A\mathbf{x}$:

Solution:

$\mathbf{x}(t) = c_1 e^{\lambda_1 t}\mathbf{v}_1 + c_2 e^{\lambda_2 t}\mathbf{v}_2 + \cdots + c_n e^{\lambda_n t}\mathbf{v}_n$

Each term corresponds to an eigenmode.

Stability Analysis

• All eigenvalues have negative real parts → stable (solutions decay)
• Any eigenvalue has positive real part → unstable (solutions grow)
• Eigenvalues on imaginary axis → oscillatory

Principal Component Analysis (PCA)

Given data covariance matrix $C$:

• Eigenvalues = variance in each principal direction
• Eigenvectors = principal directions
• Sort by eigenvalue for dimensionality reduction

$\text{Data variance along } \mathbf{v}_i = \lambda_i$

$\text{Total variance} = \lambda_1 + \lambda_2 + \cdots + \lambda_n$

Vibration Analysis

For mechanical systems:

• Eigenvalues → natural frequencies ($\omega^2 = \lambda$)
• Eigenvectors → mode shapes

Mass-spring system: $M\ddot{\mathbf{x}} = -K\mathbf{x}$

Eigenvalue problem: $K\mathbf{v} = \omega^2 M\mathbf{v}$

Image Compression

Singular Value Decomposition (SVD) uses eigenvalues:

$A = U\Sigma V^T$

Keep largest singular values (related to eigenvalues of $A^T A$) for compression.

Google PageRank

Web page importance via eigenvector of link matrix:

• Pages are nodes
• Links define transition probabilities
• Dominant eigenvector gives page rankings

Quantum Mechanics

• Eigenvalues = measurable quantities (energy, momentum)
• Eigenvectors = quantum states
• Schrödinger equation is an eigenvalue problem

$\hat{H}\psi = E\psi$

Summary of Applications

Field	Eigenvalue Interpretation
Mechanics	Natural frequencies
Circuits	Resonant frequencies
Statistics	Variance components
Graph Theory	Connectivity, PageRank
Quantum Physics	Energy levels
Control Theory	System stability
Machine Learning	Feature importance