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
| Eigenvalue | Check | Residual |
|---|---|---|
4.618034 | dominant value for [[4,1],[1,3]] | 0 after exact substitution |
2.381966 | second value for the same matrix | 0 after exact substitution |
Static sample: the characteristic polynomial is λ² - 7λ + 11 = 0. The trace check is 4.618034 + 2.381966 = 7, and the determinant check is 4.618034 × 2.381966 = 11.
Run Compute to replace this static sample with eigenvectors, multiplicities, diagonalizability, and the 2x2 visualization.
How it's calculated
- Start from
A = [[4,1],[1,3]], sotrace(A)=7anddet(A)=11. - For a 2x2 matrix, solve
λ² - trace(A)λ + det(A)=0. - The two roots are
(7±√5)/2, approximately4.618034and2.381966.
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 Eigenvalue and Eigenvector Calculator effectively
What this page is for
Use this page to inspect eigenvalues, eigenvectors, characteristic polynomial, and diagonalization clues for a matrix. Start with a small matrix and verify entries before relying on the algebraic output.
Input checks
Keep row and column order fixed when copying a matrix. Decimal entries, repeated eigenvalues, and near-singular matrices can make interpretation harder, so exact or simple values are best for a first pass.
Workflow
A practical sequence is to compute eigenvalues, review multiplicity, inspect eigenvectors, and then decide whether the result supports diagonalization or a stability interpretation.
Common mistakes
Avoid treating rounded eigenvectors as unique. Eigenvectors can be scaled, signs can flip, and repeated eigenvalues need extra checks on independent vectors.
How to read the result
Focus on whether the eigenvalues are real or complex, repeated or distinct, and how their signs or magnitudes relate to your application. That is usually more useful than the raw vector scale.