Quick start
Paste a 2x2 or 3x3 matrix and click Compute.
Read eigenvalues first, then review eigenvectors and residuals.
Turn on Teacher mode when you need more step detail.
Inputs
2x2 visualization (grid, eigenvectors, power iteration)
The 2x2 visualization appears only when Matrix A is 2x2.
Results
How it's calculated
Teacher notes
FAQ
Can I export the steps or share the setup?
Press Alt+S, or use Share, to copy a URL with current inputs. Press Alt+L to copy a LaTeX summary of the characteristic polynomial.
What happens with repeated or defective eigenvalues?
The tool compares algebraic and geometric multiplicity. If geometric multiplicity is smaller, it marks the matrix as non-diagonalizable and highlights the defective eigenspace.
What matrix should I enter first?
Start with a small 2×2 matrix when you want to confirm the workflow, then move to 3×3 only after you understand the eigenvalue pattern. This makes it easier to check whether a repeated eigenvalue or zero determinant is driving the result.
Why can two matrices give different eigenvector patterns?
Even when eigenvalues overlap, the eigenspace can change with the matrix entries. Compare multiplicity, residuals, and whether each eigenvalue has enough independent eigenvectors before deciding the matrix is diagonalizable.
How should I verify the answer?
Check that A·v and λv match for each eigenpair, then review the residuals and determinant or trace as a quick sanity check. For classroom work, compare one eigenpair by hand before moving on.
How to use Eigenvalues & Eigenvectors (2x2, 3x3) — with steps effectively
What this calculator does
This page is for estimating outcomes by changing inputs in one controlled workflow. The model keeps your focus on variables, not output shape. Start with stable assumptions, then test sensitivity by changing one key input at a time to observe directional impact.
Input meaning and unit policy
Each input has an expected unit and a typical range. For reliable interpretation, check whether you are using the same unit system, period, and base assumptions across all runs. Unit mismatch is the most common source of unexpected drift in numeric results.
Use-case sequence
A practical sequence is: first run with defaults, then create a baseline log, then run one alternative scenario, and finally compare only the changed output metric. This sequence reduces cognitive load and prevents false pattern recognition in early experiments.
Common mistakes to avoid
Avoid changing too many variables at once, mixing incompatible data sources, and interpreting a one-time output without checking robustness. A single contradictory input can flip conclusions, so keep each experiment minimal and document assumptions as part of your note.
Interpretation guidance
Review both magnitude and direction. Direction tells you whether a strategy moves outcomes in the desired direction, while magnitude helps you judge practicality. If both agree, you can proceed; if not, rebuild the baseline and verify constraints before deciding.