Eigenvalue & Eigenvector Calculator
Find eigenvalues, eigenvectors, characteristic polynomial, and diagonalisation for 2×2 and 3×3 matrices.
Characteristic Polynomial det(A − λI) = 0:
λ² − 7.00λ + 10.00 = 0
Eigenvalues:
λ1 = 5.0000
λ2 = 2.0000
Eigenvectors (normalised):
v1 = [0.707, 0.707]
v2 = [-0.447, 0.894]
Need symbolic eigenvalues, Jordan form, or matrix powers? Continue in CAS workspace →
What This Calculator Does
The eigenvalue and eigenvector calculator solves the equation Av = λv for real square matrices up to 3×3. For 2×2 matrices it uses the exact analytic formula via the characteristic polynomial; for 3×3 matrices it extracts the cubic characteristic polynomial and finds real roots by bisection, then solves (A − λI)v = 0 by Gaussian elimination to produce each eigenvector. Complex (conjugate-pair) eigenvalues are fully supported for 2×2 matrices and displayed in rectangular form a + bi.
The calculator also checks whether the matrix is diagonalisable (distinct real eigenvalues are sufficient) and, if so, displays the eigenvector matrix P and diagonal matrix D such that A = PDP⁻¹.
Formula Reference
Characteristic polynomial
The eigenvalues λ are the roots of det(A − λI) = 0. For a 2×2 matrix this gives the quadratic:
λ² − tr(A)·λ + det(A) = 0
where tr(A) = a₁₁ + a₂₂ and det(A) = a₁₁a₂₂ − a₁₂a₂₁.
Discriminant: Δ = tr(A)² − 4·det(A)
- Δ > 0: two distinct real eigenvalues
- Δ = 0: one repeated real eigenvalue (defective or not)
- Δ < 0: complex conjugate pair λ = α ± βi
Eigenvector equation
For each eigenvalue λᵢ, solve (A − λᵢI)v = 0 by row-reducing the augmented matrix. The solution is the null space of (A − λᵢI) — pick any non-zero vector in that space as the eigenvector.
Diagonalisation A = PDP⁻¹
If A has n linearly independent eigenvectors v₁, …, vₙ, form P with those columns. D is the diagonal matrix diag(λ₁, …, λₙ). Then A = PDP⁻¹ and Aᵏ = PDᵏP⁻¹, making matrix powers trivial.
Spectral properties
tr(A) = λ₁ + λ₂ + … + λₙ
det(A) = λ₁ · λ₂ · … · λₙ
These identities let you sanity-check computed eigenvalues instantly.
Worked Examples
Example 1 — 2×2, two real eigenvalues
A = [[4, 1], [2, 3]]
Characteristic polynomial: λ² − (4+3)λ + (4·3 − 1·2) = λ² − 7λ + 10 = 0
Roots: λ = (7 ± √(49−40)) / 2 = (7 ± 3) / 2 → λ₁ = 5, λ₂ = 2
Eigenvector for λ₁ = 5: (A − 5I)v = [[-1,1],[2,-2]]v = 0 → v₁ = [1, 1]
Eigenvector for λ₂ = 2: (A − 2I)v = [[2,1],[2,1]]v = 0 → v₂ = [1, -2]
Verify: tr = 5+2 = 7 ✓, det = 5×2 = 10 ✓. Matrix is diagonalisable.
Example 2 — 2×2, complex eigenvalues (rotation matrix)
A = [[0, −1], [1, 0]] (90° rotation)
Characteristic polynomial: λ² + 1 = 0
Discriminant: Δ = 0² − 4·1 = −4 < 0
Eigenvalues: λ = ±i (purely imaginary — no real fixed directions)
This is expected: a pure rotation has no real eigenvectors because no real vector keeps its direction after rotating 90°.
Example 3 — 3×3, symmetric matrix
A = [[2, 1, 0], [1, 3, 1], [0, 1, 2]]
Characteristic polynomial: −λ³ + 7λ² − 13λ + 7 = 0
Real roots (found by bisection): λ₁ ≈ 1, λ₂ ≈ 2, λ₃ ≈ 4
Symmetric matrices always have real eigenvalues and orthogonal eigenvectors (Spectral Theorem). All three eigenvalues are distinct, so A is diagonalisable.
Frequently Asked Questions
- What is an eigenvalue?
- An eigenvalue λ of a matrix A is a scalar such that Av = λv for some non-zero vector v. Geometrically, v is a direction that the linear transformation A only stretches or compresses (by factor λ), without rotating.
- What is an eigenvector?
- An eigenvector v associated with eigenvalue λ is any non-zero vector satisfying Av = λv. Eigenvectors are not unique — any scalar multiple is also an eigenvector. Calculators typically return a normalised or convenient representative.
- When does a matrix have complex eigenvalues?
- A real matrix can have complex eigenvalues if the discriminant of the characteristic polynomial is negative. They always appear in conjugate pairs a ± bi. Rotation matrices and underdamped oscillator matrices are common examples.
- What does it mean for a matrix to be diagonalisable?
- An n×n matrix is diagonalisable if it has n linearly independent eigenvectors. Sufficient conditions: all eigenvalues are distinct, or the matrix is symmetric (real). A diagonalisable matrix can be written A = PDP⁻¹, which simplifies computing Aᵏ, e^A, and solving differential equations.
- Why does this calculator only support up to 3×3?
- For 2×2 and 3×3 matrices, eigenvalues can be found analytically (quadratic and cubic formulas) or by reliable root-finding on the characteristic polynomial. For 4×4 and larger, numerical iterative methods (QR algorithm, Arnoldi iteration) are required. Use the CAS workspace for larger matrices.