Eigenvalue Calculator – Find Eigenvalues of Matrices

Calculate eigenvalues of 2×2 and 3×3 matrices with our free online eigenvalue calculator. Get step-by-step solutions using characteristic polynomials with support for complex eigenvalues.

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Examples:

Understanding Eigenvalues

Eigenvalues are fundamental to linear algebra and appear throughout science and engineering. When a matrix transforms a vector, most vectors change both direction and length. But certain special vectors – called eigenvectors – only get stretched or shrunk, never rotated. The factor by which they're stretched is the eigenvalue.

Think of eigenvalues as the "natural frequencies" of a matrix transformation. They reveal the intrinsic behavior of the system the matrix represents. In physics, eigenvalues might represent energy levels. In data science, they show the importance of principal components. In engineering, they determine system stability.

How Eigenvalue Calculation Works

1
Set up the characteristic equation
For eigenvalue λ: det(A - λI) = 0, where I is the identity matrix. This creates a polynomial equation.
2
For 2×2 matrices: solve quadratic
The characteristic polynomial is λ² - trace(A)λ + det(A) = 0. Use the quadratic formula to find both eigenvalues.
3
For 3×3 matrices: solve cubic
The characteristic polynomial is cubic: λ³ - trace(A)λ² + (sum of 2×2 minors)λ - det(A) = 0. Use Cardano's formula.
4
Handle complex eigenvalues
When the discriminant is negative, eigenvalues are complex conjugates. This happens with rotation matrices and some other transformations.

Worked Examples

Example 1: Upper Triangular 2×2

Matrix: [[3, 1], [0, 2]]

Trace: 3 + 2 = 5

Determinant: 3×2 - 1×0 = 6

Characteristic: λ² - 5λ + 6 = 0

Eigenvalues: λ₁ = 3, λ₂ = 2

For triangular matrices, eigenvalues are simply the diagonal entries. No calculation needed!

Example 2: 90° Rotation Matrix

Matrix: [[0, -1], [1, 0]]

Trace: 0 + 0 = 0

Determinant: 0×0 - (-1)×1 = 1

Characteristic: λ² + 1 = 0

Eigenvalues: λ₁ = i, λ₂ = -i

Pure rotation has no real eigenvalues – every vector changes direction. The complex eigenvalues indicate rotation.

Example 3: Symmetric 2×2

Matrix: [[2, 1], [1, 2]]

Trace: 2 + 2 = 4

Determinant: 4 - 1 = 3

Characteristic: λ² - 4λ + 3 = 0

Eigenvalues: λ₁ = 3, λ₂ = 1

Symmetric matrices always have real eigenvalues. They represent stretching along perpendicular axes.

Example 4: 3×3 Symmetric Matrix

Matrix: [[4, 1, 1], [1, 4, 1], [1, 1, 4]]

Trace: 4 + 4 + 4 = 12

Determinant: 60

Eigenvalues: λ₁ = 6, λ₂ = 3, λ₃ = 3

This matrix has a repeated eigenvalue (3). The eigenspace for λ=3 is two-dimensional.

Example 5: 3×3 Diagonal Matrix

Matrix: [[1, 0, 0], [0, 2, 0], [0, 0, 3]]

Trace: 1 + 2 + 3 = 6

Determinant: 1×2×3 = 6

Eigenvalues: λ₁ = 1, λ₂ = 2, λ₃ = 3

Diagonal matrices have their diagonal entries as eigenvalues. Each standard basis vector is an eigenvector.

Example 6: 3×3 Tridiagonal Matrix

Matrix: [[2, -1, 0], [-1, 2, -1], [0, -1, 2]]

Trace: 2 + 2 + 2 = 6

Determinant: 4

Eigenvalues: λ₁ ≈ 3.414, λ₂ = 2, λ₃ ≈ 0.586

This tridiagonal matrix appears in finite difference methods. Eigenvalues are 2 - 2cos(kπ/4) for k = 1,2,3.

Quick Fact

The word "eigenvalue" comes from German – "eigen" means "own" or "characteristic." David Hilbert introduced the term in 1904. In different languages, you'll find "valeur propre" (French, meaning "proper value"), "valor propio" (Spanish), or "固有値" (Japanese, meaning "inherent value"). All capture the idea that eigenvalues are intrinsic to the matrix.

Frequently Asked Questions

What are eigenvalues used for?

Eigenvalues appear everywhere: quantum mechanics (energy levels), structural engineering (vibration modes), Google's PageRank algorithm, principal component analysis in statistics, stability analysis of differential equations, and facial recognition systems.

Can eigenvalues be complex numbers?

Yes. Real matrices can have complex eigenvalues, which always come in conjugate pairs. Complex eigenvalues indicate rotational behavior in the transformation. Symmetric matrices always have real eigenvalues.

What's the relationship between trace and eigenvalues?

The trace (sum of diagonal elements) equals the sum of all eigenvalues. Similarly, the determinant equals the product of all eigenvalues. These provide quick checks on eigenvalue calculations.

Why do triangular matrices have diagonal entries as eigenvalues?

For a triangular matrix, det(A - λI) = (a₁₁ - λ)(a₂₂ - λ)...(aₙₙ - λ). Setting this to zero gives λ = aᵢᵢ. The triangular structure makes the characteristic polynomial factor immediately.

What does a repeated eigenvalue mean?

A repeated eigenvalue means multiple linearly independent eigenvectors share the same eigenvalue, or there's a "defect" where there aren't enough eigenvectors. This affects whether the matrix can be diagonalized.

How do eigenvalues relate to matrix powers?

If A has eigenvalue λ, then Aⁿ has eigenvalue λⁿ. This makes eigenvalues crucial for computing matrix powers and solving systems of linear differential equations.

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