MCPCalc

Eigenvalue & Eigenvector Calculator

Eigenvalues, eigenvectors, characteristic polynomial, and diagonalisation for 2×2 to 6×6 matrices. Complex eigenvalues fully supported for 2×2; real eigenvalues via QR iteration for 3×3 to 6×6.

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]

Trace = λ₁ + λ₂7.0000
Determinant = λ₁ × λ₂10.0000
Diagonalisable?Yes — distinct eigenvalues

Need symbolic eigenvalues, Jordan form, complex eigenvalues for large matrices, or Aᵏ? Continue in CAS workspace →

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What This Calculator Does

The eigenvalue and eigenvector calculator solves Av = λv for real square matrices from 2×2 up to 6×6. For 2×2 matrices it uses the exact analytic quadratic formula and fully supports complex (conjugate-pair) eigenvalues. For 3×3, it extracts the cubic characteristic polynomial and locates real roots by bisection. For 4×4, 5×5, and 6×6 matrices it applies QR iteration with Wilkinson shift — the same algorithm used by production numerical libraries — to converge on real eigenvalues, then solves (A − λᵢI)v = 0 by Gaussian elimination to produce each normalised eigenvector.

The calculator checks diagonalisability (distinct eigenvalues are sufficient) and displays the eigenvector matrix P and diagonal matrix D such that A = PDP⁻¹, enabling fast matrix powers Aᵏ = PDᵏP⁻¹.

Formula & Method

Characteristic polynomial 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
  • Δ < 0: complex conjugate pair λ = α ± βi

Eigenvector equation (A − λᵢI)v = 0

For each eigenvalue λᵢ, row-reduce the augmented matrix of (A − λᵢI) by Gaussian elimination. The null-space vector of (A − λᵢI) is the eigenvector — choose the free variable to be 1 and back-substitute.

QR iteration for 4×4–6×6 (Wilkinson shift)

Each iteration computes A − μI = QR (Gram-Schmidt), then updates A ← RQ + μI, where μ = A[n−1][n−1] is the Wilkinson shift. After enough iterations the diagonal entries converge to the eigenvalues. This is the standard algorithm behind LAPACK's dgeev and NumPy's eig.

Diagonalisation A = PDP⁻¹

If A has n linearly independent eigenvectors v₁, …, vₙ, form P with those columns. D = diag(λ₁, …, λₙ). Then Aᵏ = PDᵏP⁻¹, reducing matrix powers to scalar exponentiation.

Spectral identities (sanity checks)

tr(A) = λ₁ + λ₂ + … + λₙ

det(A) = λ₁ · λ₂ · … · λₙ

The calculator displays the trace check for all matrix sizes so you can verify convergence instantly.

Worked Examples

Example 1 — 2×2, two real eigenvalues

A = [[4, 1], [2, 3]]

Characteristic polynomial: λ² − 7λ + 10 = 0

Roots: λ = (7 ± 3) / 2 → λ₁ = 5, λ₂ = 2

v₁ = [1, 1] (normalised), v₂ = [1, −2] (normalised)

Verify: tr = 5+2 = 7 ✓, det = 5×2 = 10 ✓

Example 2 — 2×2, complex eigenvalues (rotation matrix)

A = [[0, −1], [1, 0]] (90° rotation)

Characteristic polynomial: λ² + 1 = 0, Δ = −4 < 0

Eigenvalues: λ = ±i (purely imaginary — no real fixed directions)

Example 3 — 3×3, symmetric matrix

A = [[2, 1, 0], [1, 3, 1], [0, 1, 2]]

Roots of cubic characteristic polynomial: λ₁ ≈ 1, λ₂ ≈ 2, λ₃ ≈ 4

Symmetric matrices always have real eigenvalues and orthogonal eigenvectors (Spectral Theorem).

Example 4 — 5×5 upper-triangular (QR iteration)

A = diag(11, 7, 5, 3, 2) — eigenvalues are the diagonal entries

QR iteration with Wilkinson shift: λ₁=11, λ₂=7, λ₃=5, λ₄=3, λ₅=2

Trace check: 11+7+5+3+2 = 28 = tr(A) ✓. det check: 11×7×5×3×2 = 2,310 ✓

Use the 5×5 preset above — the default matrix is exactly this diagonal example.

Example 5 — 4×4, tridiagonal symmetric matrix

A = [[5, 2, 0, 0], [2, 4, 1, 0], [0, 1, 3, 1], [0, 0, 1, 2]]

All eigenvalues are real (symmetric). QR iteration converges quickly for tridiagonal structure.

Expected eigenvalues ≈ 6.56, 4.38, 2.62, 0.44 (all real and distinct → diagonalisable).

Frequently Asked Questions

What is an eigenvalue?
An eigenvalue λ is a scalar such that Av = λv for some non-zero vector v. Geometrically, v is a direction the linear transformation A only stretches (by λ), never rotates.
What is a 4×4 eigenvalue calculator used for?
4×4 eigenvalue problems appear in structural vibration analysis (mode shapes of 4-degree-of-freedom systems), quantum mechanics (4-state systems), control theory (4th-order systems), and principal component analysis (4-feature datasets). The same QR algorithm used here is the core of MATLAB's eig and NumPy's linalg.eig.
How do I find the eigenvalues of a 5×5 matrix?
Select the 5×5 option, enter your matrix entries, and the calculator applies QR iteration with Wilkinson shift. The diagonal of the converged quasi-upper-triangular matrix gives the eigenvalues. For the 5×5 case there is no closed-form formula (Abel–Ruffini theorem: degree 5 polynomials have no general radical solution), so numerical iteration is the standard approach.
Can this calculator find 6×6 eigenvalues?
Yes. Select the 6×6 option. The QR iteration algorithm works for any size; the implementation here is validated against NumPy eig for real-distinct, repeated, symmetric, and upper-triangular test cases. Complex eigenvalues in 4×4–6×6 matrices require a more specialised treatment — use the CAS workspace for those.
What does it mean for a matrix to be diagonalisable?
A n×n matrix is diagonalisable if it has n linearly independent eigenvectors. Sufficient conditions: all eigenvalues are distinct, or the matrix is symmetric. Diagonalisability enables A = PDP⁻¹ and makes computing Aᵏ, e^A, and solving ODE systems trivial.
When should I use the CAS workspace instead?
Use the CAS workspace when you need symbolic eigenvalues (exact radical or algebraic expressions), Jordan normal form, complex eigenvalues for matrices larger than 2×2, or matrix exponential e^A.

See Also