Introduction
Eigenvalue method: fundamental tool for solving linear systems of ordinary differential equations (ODEs). Converts system into algebraic problem: find eigenvalues and eigenvectors of coefficient matrix. Enables explicit solution form, stability classification, and qualitative behavior analysis. Widely applied in physics, engineering, biology, economics.
"Eigenvalues reveal the intrinsic dynamics of linear systems, unlocking solution pathways otherwise obscured." -- Gilbert Strang
Preliminaries: Linear Systems of ODEs
Definition
System form: dX/dt = A X, where X(t) ∈ ℝⁿ vector, A ∈ ℝⁿˣⁿ constant matrix. Goal: find X(t) satisfying system.
Homogeneous Systems
Focus on homogeneous linear systems: no forcing terms. Nonhomogeneous systems treated via superposition and variation of parameters.
Matrix Notation and Properties
Matrix A assumed constant, real or complex entries. Solution space dimension equals system size n.
Eigenvalues and Eigenvectors
Definitions
Eigenvalue λ: scalar satisfying (A - λI)v = 0 for some nonzero vector v. Vector v: eigenvector corresponding to λ.
Existence
Every n×n matrix has n eigenvalues (counting multiplicities) over ℂ (Fundamental Theorem of Algebra).
Physical Interpretation
Eigenvectors: directions invariant under A. Eigenvalues: scaling factors in those directions.
Characteristic Equation
Definition
Polynomial equation det(A - λI) = 0. Roots λ are eigenvalues.
Calculation
Compute determinant symbolically or numerically. Degree equals n.
Example
For A = [[2,1],[1,2]], characteristic polynomial: (2 - λ)² - 1 = 0 → λ² - 4λ + 3 = 0.
det(A - λI) = 0General Solution Structure
Eigenbasis Decomposition
Write X(t) as linear combination of eigenvectors scaled by e^{λt}.
Formula
General solution: X(t) = Σ c_i v_i e^{λ_i t}, i = 1,...,n, with constants c_i from initial conditions.
Linearity
Solutions form vector space. Superposition principle applies.
X(t) = c_1 v_1 e^{λ_1 t} + c_2 v_2 e^{λ_2 t} + ... + c_n v_n e^{λ_n t}Diagonalization of Matrix Systems
Definition
Matrix A diagonalizable if ∃ invertible P such that P⁻¹ A P = D (diagonal matrix).
Relation to Eigenvectors
Columns of P: eigenvectors. Diagonal entries of D: eigenvalues.
Solution via Diagonalization
Change variable Y = P⁻¹ X transforms system to dY/dt = D Y, decoupled equations.
| System | Transformed System |
|---|---|
| dX/dt = A X | dY/dt = D Y, with Y = P⁻¹ X |
Complex Eigenvalues and Solutions
Occurrence
Real matrices may have complex conjugate eigenvalues λ = α ± iβ.
Solution Form
Solutions involve exponentials and trigonometric functions: e^{α t}(cos β t, sin β t).
Real-Valued Solutions
Use real and imaginary parts of complex eigenvector solutions to form real-valued solutions.
If λ = α + iβ, v = p + iq thenX(t) = e^{α t} [ (c_1 p - c_2 q) cos(β t) + (c_1 q + c_2 p) sin(β t) ]Repeated Eigenvalues and Generalized Eigenvectors
Definition
Eigenvalue λ has multiplicity > 1 but fewer than corresponding eigenvectors: defective matrix.
Generalized Eigenvectors
Solve (A - λI)^k v = 0 for k > 1 to find generalized eigenvectors.
Solution Form
Solutions include terms multiplied by t: e^{λ t} v and t e^{λ t} w.
| Eigenvalue Multiplicity | Solution Terms |
|---|---|
| Simple (1) | e^{λt} v |
| Repeated (≥2) | e^{λt} v, t e^{λt} w, ... |
For repeated λ:X(t) = e^{λ t} (v + t w)where (A - λI) v = 0, (A - λI) w = vStability Analysis via Eigenvalues
Criteria
Re(λ) < 0 for all eigenvalues → asymptotically stable (solutions decay). Re(λ) > 0 → unstable (solutions grow).
Types of Stability
Stable node, saddle point, spiral point classified by eigenvalue signs and imaginary parts.
Phase Portraits
Eigenvalues determine qualitative behavior of trajectories in phase space.
Solving Initial Value Problems
Given Initial Condition
X(0) = X₀ specified. Find constants c_i in general solution.
Procedure
Set t=0 in general solution, solve linear system for c_i using eigenvectors.
Uniqueness and Existence
Solution unique for each initial vector X₀ due to linearity and invertible eigenvector matrix (if diagonalizable).
Given X(0) = X₀,X(0) = Σ c_i v_i = V c = X₀Solve for c: c = V⁻¹ X₀Examples
Example 1: Distinct Real Eigenvalues
System: dX/dt = [[3,1],[0,2]] X. Eigenvalues λ₁=3, λ₂=2. Eigenvectors v₁=[1,0], v₂=[1,1].
Example 2: Complex Eigenvalues
System: dX/dt = [[0,-1],[1,0]] X. Eigenvalues λ= ±i. Solutions oscillatory.
Example 3: Repeated Eigenvalues
System: dX/dt = [[2,1],[0,2]] X. Eigenvalue λ=2 (multiplicity 2), one eigenvector. Generalized eigenvector used.
Limitations and Extensions
Nonlinear Systems
Eigenvalue method applies only to linear or linearized systems near equilibria.
Non-Diagonalizable Matrices
Requires generalized eigenvectors and Jordan normal form.
Numerical Computation
Computational methods needed for large or complicated matrices (QR algorithm, power iteration).
References
- H. K. Khalil, Nonlinear Systems, 3rd ed., Prentice Hall, 2002, pp. 85-120.
- G. Strang, Introduction to Linear Algebra, 5th ed., Wellesley-Cambridge Press, 2016, pp. 345-370.
- W. E. Boyce and R. C. DiPrima, Elementary Differential Equations and Boundary Value Problems, 10th ed., Wiley, 2012, pp. 245-280.
- S. H. Strogatz, Nonlinear Dynamics and Chaos, 2nd ed., Westview Press, 2015, pp. 50-75.
- L. Perko, Differential Equations and Dynamical Systems, 3rd ed., Springer, 2001, pp. 100-135.