Eigenvalue Calculator

What are Eigenvalues and Eigenvectors?

Eigenvalues (λ) and Eigenvectors (v) are fundamental concepts in linear algebra. For a square matrix A, an eigenvector v is a nonzero vector that, when multiplied by A, yields a scalar multiple of itself: Av = λv.

The eigenvalue λ represents the factor by which the eigenvector is scaled during the transformation. Eigenvalues reveal important properties of matrices, including stability, oscillatory behavior, and principal directions in data.

Eigenvalue Formulas for 2×2 Matrices

Step-by-Step Calculation Example

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

1. Given matrix: A = [[2, 1], [1, 2]]
2. Trace: tr(A) = 2 + 2 = 4
3. Determinant: det(A) = (2×2) - (1×1) = 3
4. Characteristic polynomial: λ² - 4λ + 3 = 0
5. Solve quadratic: (λ - 3)(λ - 1) = 0
6. Eigenvalues: λ₁ = 3, λ₂ = 1
7. Eigenvector for λ₁=3: Solve (A-3I)v=0 → [[-1,1],[1,-1]]v=0 → v₁ = [1,1]ᵀ
8. Eigenvector for λ₂=1: Solve (A-I)v=0 → [[1,1],[1,1]]v=0 → v₂ = [1,-1]ᵀ

Types of Eigenvalues

1. Real and Distinct

When discriminant D > 0:

• Two different real eigenvalues
• Example: [[2,1],[1,2]] → λ=3, λ=1
• Matrix is diagonalizable
• Eigenvectors are linearly independent

2. Real and Repeated

When discriminant D = 0:

• One repeated eigenvalue
• Example: [[1,0],[0,1]] → λ=1, λ=1
• May or may not be diagonalizable
• Check geometric multiplicity

3. Complex Conjugate

When discriminant D < 0:

• Complex eigenvalues in conjugate pairs
• Example: [[0,-1],[1,0]] → λ=i, λ=-i
• Associated with rotations
• Real matrices have complex conjugate eigenvalues

Properties of Eigenvalues

Property	Formula	Description	Example
Trace Sum	λ₁ + λ₂ = a + d	Sum of eigenvalues equals trace	For [[2,1],[1,2]]: 3+1=4
Determinant Product	λ₁ × λ₂ = ad - bc	Product equals determinant	3×1=3
Real Symmetric	All λ real	Symmetric matrices have real eigenvalues	[[2,1],[1,2]] has real λ
Orthogonal	|λ| = 1	Eigenvalues on unit circle	Rotation matrices

Applications of Eigenvalues

Physics & Engineering

• Vibration analysis: Natural frequencies as eigenvalues
• Quantum mechanics: Energy levels as eigenvalues of Hamiltonian
• Stability analysis: System stability determined by eigenvalues
• Structural mechanics: Buckling loads as eigenvalues

Data Science & Machine Learning

• Principal Component Analysis (PCA): Eigenvalues indicate variance
• PageRank algorithm: Eigenvector of web link matrix
• Image compression: Singular value decomposition uses eigenvalues
• Recommendation systems: Matrix factorization techniques

Economics & Finance

• Portfolio optimization: Eigenvalues of covariance matrix
• Input-output models: Economic growth rates as eigenvalues
• Risk analysis: Eigenvalues measure portfolio risk

Common Examples

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Frequently Asked Questions (FAQs)

Q: What's the difference between eigenvalues and eigenvectors?

A: Eigenvalues (λ) are scalars that represent the scaling factor. Eigenvectors (v) are vectors that don't change direction when transformed by the matrix. They satisfy the equation Av = λv.

Q: Can a matrix have complex eigenvalues?

A: Yes! Real matrices can have complex eigenvalues, which always appear in conjugate pairs (a+bi and a-bi). This happens when the discriminant D = tr(A)² - 4det(A) is negative.

Q: How do I find eigenvectors from eigenvalues?

A: For each eigenvalue λ, solve the homogeneous system (A - λI)v = 0. The nonzero solutions form the eigenspace for that λ. Use Gaussian elimination to find basis vectors.

Q: What does it mean if eigenvalues are repeated?

A: Repeated eigenvalues (algebraic multiplicity > 1) may or may not have enough linearly independent eigenvectors. If geometric multiplicity < algebraic multiplicity, the matrix is defective and not diagonalizable.

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