The Eigenvalue Calculator computes eigenvalues and eigenvectors for square matrices. These fundamental concepts in linear algebra are essential for understanding matrix transformations, solving differential equations, analyzing stability in systems, and many applications in physics, engineering, and data science.

Key Features

• Eigenvalue Calculation: Find all eigenvalues λ that satisfy the equation Av = λv for a given matrix A.
• Eigenvector Computation: Calculate the corresponding eigenvectors for each eigenvalue.
• Characteristic Polynomial: Display the characteristic polynomial det(A - λI) = 0 used to find eigenvalues.
• Matrix Sizes: Support for 2×2, 3×3, and larger square matrices.
• Complex Eigenvalues: Handle matrices with complex eigenvalues and eigenvectors.
• Diagonalization: Determine if a matrix is diagonalizable and compute the diagonal form.

What are Eigenvalues and Eigenvectors?

For a square matrix A, an eigenvector v is a non-zero vector that, when multiplied by A, produces a scalar multiple of itself. That scalar is the eigenvalue λ:

In other words, the transformation represented by A stretches or shrinks the eigenvector v by a factor of λ, but doesn't change its direction (except possibly reversing it if λ is negative).

Example 1: 2×2 Matrix

Find eigenvalues and eigenvectors for:

A = [4  1]
    [2  3]

Step 1: Find characteristic polynomial
det(A - λI) = det([4-λ   1  ]) = (4-λ)(3-λ) - 2 = 0
                 [2    3-λ]

Step 2: Expand and solve
λ<sup>2</sup> - 7λ + 10 = 0
(λ - 5)(λ - 2) = 0
Eigenvalues: λ<sub>1</sub> = 5, λ<sub>2</sub> = 2

Step 3: Find eigenvectors
For λ<sub>1</sub> = 5:
(A - 5I)v = 0
[-1  1][v<sub>1</sub>] = [0]
[ 2 -2][v<sub>2</sub>]   [0]
Eigenvector: v<sub>1</sub> = [1, 1]ᵀ

For λ<sub>2</sub> = 2:
(A - 2I)v = 0
[2  1][v<sub>1</sub>] = [0]
[2  1][v<sub>2</sub>]   [0]
Eigenvector: v<sub>2</sub> = [1, -2]ᵀ

Example 2: Identity Matrix

Find eigenvalues of the 2×2 identity matrix:

I = [1  0]
    [0  1]

For any vector v: Iv = 1·v

Therefore:
Eigenvalue: λ = 1 (with multiplicity 2)
Eigenvectors: Any non-zero vector in ℝ<sup>2</sup>

This means the identity transformation doesn't change
any vector's direction or magnitude.

Example 3: Rotation Matrix

Find eigenvalues for a 90° rotation matrix:

R = [0  -1]
    [1   0]

Characteristic polynomial:
det(R - λI) = λ<sup>2</sup> + 1 = 0
λ<sup>2</sup> = -1
Eigenvalues: λ = ±i (complex numbers)

This indicates that rotation has no real eigenvectors,
as no vector maintains its direction under rotation.

Example 4: Diagonal Matrix

Find eigenvalues of a diagonal matrix:

D = [3  0  0]
    [0  5  0]
    [0  0  2]

For diagonal matrices, eigenvalues are the diagonal entries:
Eigenvalues: λ<sub>1</sub> = 3, λ<sub>2</sub> = 5, λ<sub>3</sub> = 2

Eigenvectors: Standard basis vectors e<sub>1</sub>, e<sub>2</sub>, e<sub>3</sub>

• Characteristic Equation: Always start by computing det(A - λI) = 0 to find eigenvalues.
• Sum and Product: For a 2×2 matrix, the sum of eigenvalues equals the trace (sum of diagonal elements), and their product equals the determinant.
• Zero Eigenvalue: If λ = 0 is an eigenvalue, the matrix is singular (non-invertible).
• Real Symmetric Matrices: Always have real eigenvalues and orthogonal eigenvectors.
• Complex Eigenvalues: For real matrices, complex eigenvalues always come in conjugate pairs (a ± bi).
• Eigenvector Scaling: Eigenvectors are not unique; any scalar multiple of an eigenvector is also an eigenvector.
• Geometric Interpretation: Eigenvalues represent scaling factors along the directions of their corresponding eigenvectors.
• Verification: Always check your answer by computing Av and λv to ensure they're equal.