Eigenvalue visualizer
Find the exact characteristic polynomial of a square matrix of fractions. Rational and real quadratic eigenvalues include exact eigenvectors; higher-degree real roots are isolated exactly, and higher-degree complex roots are numerical approximations.
Square matrices from 1 by 1 to 8 by 8, up to 20,000 characters, with exact entries of at most 100 digits; factoring uses factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of eigenvalues and an SVG plot. One row per line or separated by semicolons; entries such as 3, -0.5 or 2/3. Up to 20,000 characters. Review the settings before running.
Your result
How to use Eigenvalue visualizer
- Paste a square matrix, one row per line, such as 4 1 and 2 3.
- Run the visualizer to compute the characteristic polynomial and factor it.
- Read the eigenvalue kinds, multiplicities, supported eigenvectors and diagonalizability status. A null status means the method did not determine it.
- Download the JSON report, the CSV of eigenvalues or the plot.
Useful next steps
Square matrices from 1 by 1 to 8 by 8, up to 20,000 characters, with exact entries of at most 100 digits; factoring uses factors up to degree 24. Everything is computed on your device in a background worker with a 15 second limit. Downloads are a JSON report, a CSV of eigenvalues and an SVG plot.
Exact method
The characteristic polynomial det(tI - A) is computed with the Faddeev-LeVerrier recurrence in exact fractions and checked against the determinant. It is factored over the rationals. Rational and real quadratic roots give exact eigenvectors from the null space of A - tI, verified by multiplication. Eigenvectors for other root kinds are not computed. Method: Strang, Introduction to Linear Algebra, 5th edition, chapter 6 (2016); Faddeev and Faddeeva, Computational Methods of Linear Algebra (1963).
Higher-degree factors
Real roots of irreducible cubic or higher factors are isolated exactly with Sturm sequences and shown with a decimal value. Complex roots of those factors are numerical approximations with polynomial residuals. A small residual checks a candidate root; it does not prove that every complex root was found or that a complete eigenbasis was computed.
Plot
For a 2 by 2 matrix with real eigenvalues the plot shows the unit circle, its image and the invariant directions. Otherwise it shows the eigenvalues in the complex plane.