What matrix sizes does this tool support?

This tool supports 2x2 matrices. For a 2x2 matrix the characteristic polynomial is a simple quadratic that can be solved exactly, making it ideal for showing every step clearly.

What if the discriminant is negative?

If the discriminant (b² − 4ac of the characteristic polynomial) is negative, the matrix has two complex conjugate eigenvalues. The tool displays them in a + bi form and shows the complex eigenvectors.

Are the eigenvectors normalized?

Eigenvectors are displayed as simplified direction vectors (e.g. [1, k]) rather than unit vectors. You can normalize them yourself by dividing each component by the vector's magnitude.

What does it mean if both eigenvalues are the same?

A repeated eigenvalue (degenerate case) means the matrix may have only one independent eigenvector. The tool flags this and still computes the single eigenvector direction when one exists.

Can I use this for non-square entries like fractions?

Yes. Enter decimal equivalents (e.g. 0.5 instead of 1/2). The tool performs floating-point arithmetic and rounds display values to 6 significant figures.

Eigenvector Calculator

Name: Eigenvector Calculator
Availability: InStock
Author: Nham Vu

Enter a 2x2 matrix and instantly compute its eigenvalues and eigenvectors step by step.

Enter 2×2 Matrix

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Example Matrices

Enter matrix values and click Calculate

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Summary

Enter a 2x2 matrix and instantly compute its eigenvalues and eigenvectors step by step.

How it works

Enter the four entries of your 2x2 matrix (a, b, c, d).
The tool forms the characteristic polynomial: λ² − (a+d)λ + (ad−bc) = 0.
It solves the quadratic to find eigenvalues λ₁ and λ₂ (real or complex).
For each eigenvalue, it solves (A − λI)v = 0 to find the eigenvector direction.
Results are displayed with full step-by-step working you can copy or verify.

Use cases

Check linear algebra homework involving 2x2 eigenvalue problems.
Verify hand-computed eigenvectors before submitting coursework.
Understand principal component directions in small covariance matrices.
Analyze stability of 2D dynamical systems by inspecting eigenvalue signs.
Find the axes of rotation or scaling for 2D transformation matrices.
Confirm diagonalizability by checking whether two distinct eigenvectors exist.
Prototype eigenvalue algorithms before scaling to larger matrices.
Quickly decompose simple Markov chain transition matrices.

Frequently Asked Questions

Last updated: 2026-06-10 · Reviewed by Nham Vu