Eigenvalue Calculator

Calculate eigenvalues and eigenvectors for 2×2 and 3×3 matrices with step-by-step solutions

2×2 Matrix 3×3 Matrix

Enter matrix elements:

What are Eigenvalues and Eigenvectors?

Eigenvalues and eigenvectors are fundamental concepts in linear algebra that describe how a matrix transformation affects certain vectors. When a matrix A is multiplied by its eigenvector v, the result is simply the eigenvector scaled by a constant factor λ (lambda), which is the eigenvalue: Av = λv.

Key Properties

Eigenvalue (λ): A scalar value that represents how much the eigenvector is scaled during matrix transformation
Eigenvector (v): A non-zero vector that maintains its direction when transformed by the matrix
Characteristic Polynomial: Obtained from det(A – λI) = 0, where I is the identity matrix
Multiplicity: The number of times an eigenvalue appears as a root of the characteristic polynomial

Applications

Principal Component Analysis (PCA): Data dimensionality reduction and pattern recognition
Stability Analysis: Determining system stability in engineering and physics
Quantum Mechanics: Finding energy states and wave functions
Computer Graphics: 3D transformations and animations
Vibration Analysis: Natural frequencies in mechanical systems

How to Calculate Eigenvalues

The process involves several systematic steps that transform the original matrix problem into a polynomial equation.

Step-by-Step Method

Form the characteristic matrix: Subtract λ from each diagonal element to get (A – λI)
Calculate the determinant: Find det(A – λI) to obtain the characteristic polynomial
Solve the characteristic equation: Set det(A – λI) = 0 and solve for λ values
Find eigenvectors: For each eigenvalue, solve (A – λI)v = 0 to get corresponding eigenvectors

Example Calculation

For matrix: [2, 1; 0, 3]

Characteristic polynomial: (2-λ)(3-λ) = 0

Eigenvalues: λ₁ = 2, λ₂ = 3

Frequently Asked Questions

Can a matrix have complex eigenvalues?

Yes, real matrices can have complex eigenvalues. They always appear in conjugate pairs (a + bi, a – bi) when the matrix has real entries.

What happens if eigenvalues are repeated?

Repeated eigenvalues indicate geometric multiplicity. The matrix may have fewer linearly independent eigenvectors than its size, affecting diagonalisation.

Why might eigenvalues be zero?

Zero eigenvalues indicate that the matrix is singular (non-invertible) and has a non-trivial null space. This means some vectors are mapped to the zero vector.

How do I interpret negative eigenvalues?

Negative eigenvalues indicate that the corresponding eigenvectors are reversed in direction during transformation, whilst being scaled by the absolute value.

Are eigenvectors unique?

Eigenvectors are unique up to scalar multiplication. Any non-zero scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.

Matrix Types and Their Eigenvalue Properties

Symmetric Matrices

Symmetric matrices (where A = Aᵀ) always have real eigenvalues and orthogonal eigenvectors. This property makes them particularly useful in applications like PCA and quadratic forms.

Diagonal Matrices

For diagonal matrices, the eigenvalues are simply the diagonal elements, and the eigenvectors are the standard basis vectors. This makes eigenvalue computation trivial.

Triangular Matrices

Upper and lower triangular matrices have eigenvalues equal to their diagonal elements. The eigenvector calculation requires back-substitution methods.

Orthogonal Matrices

Orthogonal matrices have eigenvalues with absolute value 1. They represent rotations and reflections, preserving lengths and angles in transformations.