Eigenvector Calculator

What are Eigenvectors and Eigenvalues?

Eigenvectors and eigenvalues are fundamental concepts in linear algebra. For a square matrix A, an eigenvector v is a non-zero vector that, when multiplied by A, only changes by a scalar factor λ (the eigenvalue). The relationship is expressed as:

Key Properties of Eigenvectors

Finding Eigenvectors: Step-by-Step

1. Find Eigenvalues

Solve the characteristic equation:

det(A - λI) = 0
For 2×2: λ² - tr(A)λ + det(A) = 0
Example: A = [[2,1],[1,2]] → λ² - 4λ + 3 = 0
Solutions: λ₁ = 3, λ₂ = 1

2. Find Eigenvectors for Each λ

Solve (A - λI)v = 0:

For λ₁ = 3: (A - 3I)v = [[-1,1],[1,-1]]v = 0
Solve: -v₁ + v₂ = 0 → v₁ = v₂
Eigenvector: v₁ = [1, 1]ᵀ (or any scalar multiple)
For λ₂ = 1: v₂ = [1, -1]ᵀ

3. Verify Solution

Check A·v = λ·v:

A·v₁ = [[2,1],[1,2]]·[1,1]ᵀ = [3,3]ᵀ = 3·[1,1]ᵀ ✓
A·v₂ = [[2,1],[1,2]]·[1,-1]ᵀ = [1,-1]ᵀ = 1·[1,-1]ᵀ ✓

Types of Eigenvalues

Type	Properties	Example Matrix	Eigenvalues	Applications
Real & Distinct	All eigenvalues real and different	[[2,1],[1,2]]	3, 1	Most common case
Complex	Complex eigenvalues in conjugate pairs	[[0,-1],[1,0]]	i, -i	Rotation matrices
Repeated	Same eigenvalue multiple times	[[2,0],[0,2]]	2 (multiplicity 2)	Scalar multiples of identity
Zero Eigenvalue	λ = 0 means matrix is singular	[[1,1],[1,1]]	0, 2	Rank-deficient matrices

Common Matrices and Their Eigenvalues

Matrix Type	2×2 Example	Eigenvalues	Eigenvectors	Properties
Symmetric	[[a,b],[b,a]]	a±b	[1,1]ᵀ, [1,-1]ᵀ	Real eigenvalues, orthogonal eigenvectors
Rotation	[[cosθ,-sinθ],[sinθ,cosθ]]	e^(±iθ)	Complex vectors	Complex eigenvalues, magnitude 1
Diagonal	[[a,0],[0,b]]	a, b	[1,0]ᵀ, [0,1]ᵀ	Standard basis vectors
Triangular	[[a,b],[0,c]]	a, c	Depends on b	Eigenvalues on diagonal

Real-World Applications

Physics & Engineering

• Quantum mechanics: Energy states as eigenvalues of Hamiltonian operator
• Structural analysis: Natural frequencies as eigenvalues in vibration analysis
• Control systems: System stability determined by eigenvalues
• Fluid dynamics: Turbulence analysis using eigenmodes

Computer Science & Data Analysis

• Principal Component Analysis (PCA): Eigenvectors of covariance matrix for dimensionality reduction
• PageRank algorithm: Google's ranking using eigenvector of web graph
• Image compression: Singular Value Decomposition (SVD) uses eigenvectors
• Machine learning: Feature extraction and data transformation

Mathematics & Statistics

• Markov chains: Steady-state distribution as eigenvector with λ=1
• Differential equations: Solving systems using eigenvector methods
• Graph theory: Spectral graph theory using adjacency matrix eigenvalues
• Optimization: Hessian matrix eigenvalues determine curvature

Everyday Applications

• Facial recognition: Eigenfaces for face detection and recognition
• Recommendation systems: Collaborative filtering using SVD
• Risk analysis: Portfolio optimization in finance
• Signal processing: Filter design and noise reduction

Eigenvalue Properties

Property	Formula/Statement	Example	Significance
Trace Relationship	Sum of eigenvalues = trace(A)	λ₁ + λ₂ = a₁₁ + a₂₂	Quick eigenvalue sum check
Determinant Relationship	Product of eigenvalues = det(A)	λ₁ × λ₂ = det(A)	Quick eigenvalue product check
Spectral Radius	ρ(A) = max|λᵢ|	Largest eigenvalue magnitude	Convergence analysis
Cayley-Hamilton Theorem	p(A) = 0 where p(λ)=det(A-λI)	A satisfies its own characteristic equation	Matrix function evaluation

Step-by-Step Eigenvector Calculation

Example: Matrix A = [[2, 1], [1, 2]]

1. Find eigenvalues: Solve det(A - λI) = 0
2. det([[2-λ, 1], [1, 2-λ]]) = (2-λ)² - 1 = λ² - 4λ + 3 = 0
3. Solve: λ₁ = 3, λ₂ = 1
4. For λ₁ = 3: Solve (A - 3I)v = 0
5. [[-1, 1], [1, -1]]v = 0 → -v₁ + v₂ = 0 → v₁ = v₂
6. Eigenvector: v₁ = [1, 1]ᵀ (or any multiple)
7. For λ₂ = 1: Solve (A - I)v = 0
8. [[1, 1], [1, 1]]v = 0 → v₁ + v₂ = 0 → v₁ = -v₂
9. Eigenvector: v₂ = [1, -1]ᵀ (or any multiple)
10. Verify: A·v₁ = [3,3]ᵀ = 3v₁ ✓, A·v₂ = [1,-1]ᵀ = 1v₂ ✓

Special Cases and Important Notes

Case	Description	Example	Eigenvalue Behavior	Eigenvector Behavior
Diagonalizable	n independent eigenvectors	[[2,0],[0,3]]	Real, may repeat	Full set exists
Non-diagonalizable	Deficient eigenvectors	[[1,1],[0,1]]	Repeated (λ=1)	Only one independent eigenvector
Orthogonal Matrix	AᵀA = I	Rotation matrix	|λ| = 1	Orthonormal eigenvectors
Nilpotent Matrix	Aᵏ = 0 for some k	[[0,1],[0,0]]	All eigenvalues 0	May be deficient

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Frequently Asked Questions (FAQs)

Q: What does "eigen" mean?

A: "Eigen" is a German word meaning "own," "characteristic," or "proper." In mathematics, eigenvectors are the "characteristic vectors" of a matrix that don't change direction under the transformation represented by the matrix.

Q: Can a matrix have complex eigenvalues?

A: Yes! Real matrices can have complex eigenvalues, which always come in conjugate pairs. For example, rotation matrices have complex eigenvalues e^(±iθ). The corresponding eigenvectors are also complex.

Q: What is eigenvalue multiplicity?

A: Multiplicity refers to how many times an eigenvalue appears. Algebraic multiplicity is the number of times it's a root of the characteristic polynomial. Geometric multiplicity is the number of linearly independent eigenvectors for that eigenvalue.

Q: Why are eigenvectors important in PCA?

A: In Principal Component Analysis, eigenvectors of the covariance matrix point in directions of maximum variance in the data. The corresponding eigenvalues indicate how much variance is captured by each principal component.

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