Eigenvector Calculator

Summary

This eigenvector calculator finds the eigenvalues and eigenvectors of any 2x2 real matrix using the characteristic polynomial det(A - λI) = 0. Enter the four matrix entries and the tool solves the resulting quadratic, then computes the eigenvector for each eigenvalue by solving (A - λI)v = 0. Results include step-by-step working so you can follow along or check your hand calculations.

How it works

1. Enter the four entries of your 2x2 matrix (a, b, c, d).
2. The tool forms the characteristic polynomial: λ² − (a+d)λ + (ad−bc) = 0.
3. It solves the quadratic to find eigenvalues λ₁ and λ₂ (real or complex).
4. For each eigenvalue, it solves (A − λI)v = 0 to find the eigenvector direction.
5. 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