Complex Eigenvector Calculator

Calculator Inputs

Enter the real and imaginary parts for each matrix entry of a 2 × 2 complex matrix.

a₁₁ real

a₁₁ imaginary

a₁₂ real

a₁₂ imaginary

a₂₁ real

a₂₁ imaginary

a₂₂ real

a₂₂ imaginary

Decimal precision

Vector output mode

Quick matrix preview

A = [[a₁₁, a₁₂], [a₂₁, a₂₂]]

Default example:

[[1 + i, 1], [0, 2 - i]]

Reset

Example Data Table

Example Matrix	Expected Eigenvalues	Expected Eigenvectors	Why It Works
[[1 + i, 1], [0, 2 - i]]	λ₁ = 1 + i, λ₂ = 2 - i	v₁ = [1; 0], v₂ = [1; 1 - 2i]	Upper triangular matrices use diagonal entries as eigenvalues.
[[3, 0], [0, 3]]	λ₁ = λ₂ = 3	Any nonzero vector	A scalar identity multiple makes every direction an eigenvector.
[[2, i], [-i, 2]]	λ₁ = 1, λ₂ = 3	Computed numerically by the page	Complex off-diagonal entries still fit the same characteristic formula.

Formula Used

For a 2 × 2 complex matrix

A = [[a, b], [c, d]]

Characteristic polynomial

det(A − λI) = 0

λ² − tr(A)λ + det(A) = 0

Trace and determinant

tr(A) = a + d

det(A) = ad − bc

Eigenvalues

λ = (tr(A) ± √(tr(A)² − 4det(A))) / 2

Eigenvectors

Solve (A − λI)v = 0 for a nonzero vector v.

This page forms a representative eigenvector from one row, then optionally normalizes it to unit length.

Residual check

Residual = ‖(A − λI)v‖

Smaller residuals confirm a numerically consistent eigenpair.

How to Use This Calculator

Enter the real and imaginary parts for all four matrix entries.
Choose the decimal precision you want in the final display.
Select whether the eigenvectors should remain raw or be normalized.
Press Compute Eigenvectors to show results above the form.
Review eigenvalues, eigenvectors, residual checks, sensitivity hints, and the graph.
Use the CSV or PDF buttons to export the current result set.

Frequently Asked Questions

1) What does this calculator solve?

It finds eigenvalues and eigenvectors for a 2 × 2 matrix whose entries can contain real and imaginary parts. It also reports residual norms, separation between eigenvalues, and a practical conditioning hint.

2) Why are complex eigenvectors important?

Complex eigenvectors appear in vibration analysis, control systems, quantum mechanics, signal processing, and stability work. They help describe directions preserved by a matrix, even when rotation and oscillation are involved.

3) Why can the same eigenvector look different?

Any nonzero scalar multiple of an eigenvector represents the same eigendirection. That is why this page can show raw representative vectors or unit-normalized versions without changing the mathematical meaning.

4) What does the residual norm tell me?

The residual norm measures how closely the reported pair satisfies (A − λI)v = 0. A value near zero means the result is numerically consistent with the matrix and computed eigenvalue.

5) What if the eigenvalues are repeated?

Repeated eigenvalues may indicate either infinitely many eigenvectors or only one main eigendirection. The page checks for that pattern and reports a diagonalization note to help interpret the result.

6) Does this work for non-diagonalizable matrices?

Yes. The calculator still returns the repeated eigenvalue and a representative eigenvector. When the matrix is likely defective, the diagonalization note warns that there may be only one independent eigendirection.

7) Why is eigenvalue separation useful?

When eigenvalues are very close, small changes in the matrix can move the eigenvectors noticeably. The separation and sensitivity hint help you judge whether the result is stable or delicate.

8) Can I export the result?

Yes. After computing, use the CSV button for spreadsheet-friendly output or the PDF button for a concise printable summary. Both export the values currently shown on the page.