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Calculator Answers

Eigenvector Calculator

Blake Boege
Written by Blake Boege · Founder, Calculator Answers

An eigenvector calculator is a linear algebra tool that computes the eigenvalues and corresponding eigenvectors of a square matrix. Eigenvectors are non-zero vectors that change only by a scalar factor (the eigenvalue) when a linear transformation is applied to them. The calculator solves the characteristic equation det(A - λI) = 0 to find eigenvalues, and then computes the null space for each eigenvalue to resolve the eigenvectors. Computer scientists, engineers, and mathematicians use this calculator to solve systems of differential equations, perform principal component analysis, and analyze stability in physics models.

Enter the four entries of a 2×2 matrix. The calculator returns the trace, determinant, discriminant, eigenvalues, and the matching unit eigenvectors. Detects complex eigenvalues and reports them clearly.

See also: Matrix Calculator and Eigenvalue Calculator.

Quick Answer

Find the eigenvalues and eigenvectors of a 2x2 or 3x3 matrix. Enter the matrix coefficients to see step-by-step algebraic calculations.

2 × 2 matrix A

Enter the entries. The calculator finds eigenvalues and unit eigenvectors directly from the 2×2 characteristic polynomial.

e.g. 4

e.g. 1

e.g. 2

e.g. 3

Scope

This calculator handles real 2×2 matrices. Eigenvectors are returned as unit vectors. Any scalar multiple of an eigenvector is also an eigenvector for the same eigenvalue.

Complex eigenvalues are detected and reported, but complex eigenvectors are out of scope here. For larger matrices and general eigen-decomposition, use a dedicated linear algebra tool.

Eigenvalues and eigenvectors

Two eigenvalues

λ₁ = 5, λ₂ = 2

Eigenvectors v₁ = (-0.7071, -0.7071), v₂ = (-0.4472, 0.8944)

Trace (a + d)7
Determinant (ad − bc)10
Discriminant (trace² − 4·det)9
Eigenvalue 15
Eigenvector 1 (unit)(-0.7071, -0.7071)
Eigenvalue 22
Eigenvector 2 (unit)(-0.4472, 0.8944)

Each eigenvalue λ solves det(A − λI) = 0. The matching eigenvector v satisfies (A − λI)v = 0 and is unique up to scalar multiplication; the calculator returns the unit form.

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Examples

[[4, 1], [2, 3]]

λ₁ ≈ 5, λ₂ ≈ 2 · v₁ ≈ (0.707, 0.707), v₂ ≈ (-0.447, 0.894)

[[2, 0], [0, 5]]

λ₁ = 2, λ₂ = 5 · v₁ = (1, 0), v₂ = (0, 1)

[[3, 1], [0, 3]]

λ = 3 (repeated) · v = (1, 0)

[[0, -1], [1, 0]]

Complex λ = ±i (no real eigenvectors)

How it works

Eigenvalues come from the characteristic equation; eigenvectors come from solving the resulting null-space equation for each eigenvalue.

Characteristic eq. · det(A − λI) = λ² − trace · λ + det = 0

Discriminant · D = trace² − 4 · det

Eigenvalues · λ = (trace ± √D) / 2

Eigenvector · solve (A − λI) v = 0; normalize to unit length

D > 0: two distinct real eigenvalues. D = 0: one repeated. D < 0: complex conjugate pair (out of scope).

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Matrix CalculatorRun matrix operations on 2x2 and 3x3 matrices: addition, subtraction, scalar multiplication, matrix multiplication, transpose, determinant, and inverse. Not a system solver (see RREF and system of equations).Eigenvalue CalculatorFind the eigenvalues of a 2x2 matrix using trace and determinant. Returns the characteristic polynomial, discriminant, and real or complex eigenvalues from λ² − tr(A)·λ + det(A) = 0.

Frequently asked questions

For a square matrix A, an eigenvector is a non-zero vector v such that Av = λv for some scalar λ (the eigenvalue). Multiplying by A only stretches v by λ; it does not change v's direction. Eigenvectors describe the matrix's natural axes.

First find the eigenvalues by solving det(A − λI) = 0 (the characteristic equation). Then for each eigenvalue λ, solve the linear system (A − λI)v = 0 for v. The solution is a one-dimensional subspace; pick any non-zero vector in it. The calculator returns the unit-length form.

For A = [[a, b], [c, d]], the characteristic equation is λ² − (a+d)λ + (ad−bc) = 0. That is λ² − trace(A) · λ + det(A) = 0. The discriminant is trace² − 4·det, which decides whether eigenvalues are real and distinct, repeated, or a complex conjugate pair.

If Av = λv, then A(kv) = λ(kv) for any non-zero scalar k. So any non-zero multiple of an eigenvector is also an eigenvector. To pick one specific representative, conventions usually use the unit eigenvector (length 1) or set one coordinate to 1.

Then the matrix has a complex conjugate pair of eigenvalues. The corresponding eigenvectors are also complex-valued. This calculator reports the complex eigenvalues but does not produce complex eigenvectors. Use a dedicated linear-algebra package (NumPy, SymPy, MATLAB, etc.) for the complex case.

The matrix has a single repeated eigenvalue. There is at most a one-dimensional eigenspace for it; the calculator returns one eigenvector for the repeated eigenvalue. The matrix may also be defective (no second linearly independent eigenvector), but that distinction needs the Jordan form, not just the characteristic polynomial.

No. This calculator is scoped to 2×2 real matrices. For 3×3 and larger, the characteristic polynomial becomes high-degree and root-finding requires numerical methods (QR algorithm, power iteration, etc.). Use a CAS or scientific library for those.