Matrix input panel
Enter a square matrix, compute eigenvalues, estimate eigenvalue condition numbers, and inspect spectral sensitivity with downloadable outputs.
Plotly graphs
The left plot places eigenvalues on the complex plane. The right plot compares their estimated condition numbers.
Example data table
This symmetric example is useful for validation because normal matrices typically return eigenvalue condition numbers near 1.
| Example matrix | Expected eigenvalues | Expected condition numbers | Interpretation |
|---|---|---|---|
| [[4, 1], [1, 3]] | ≈ 4.6180, 2.3820 | ≈ 1.0000, 1.0000 | Symmetric matrices are normal, so each eigenvalue is comparatively well conditioned. |
| [[4, 2, 0], [0, 3, 1], [0, 0, 1]] | 4, 3, 1 | Greater than 1 | Upper triangular, non-normal structure usually increases sensitivity under perturbations. |
Formula used
Right eigenvector relation: A x = λ x
Left eigenvector relation: yT A = λ yT
Eigenvalue condition number: κ(λ) = ||x||2 ||y||2 / |yH x|
First-order perturbation idea: |δλ| is bounded in proportion to κ(λ) and the size of the matrix perturbation.
A condition number close to 1 suggests a stable eigenvalue. Large values indicate that small perturbations in the matrix can trigger comparatively large changes in that eigenvalue.
This calculator estimates left eigenvectors from AT and pairs them numerically with the eigenvalues of A. Repeated or nearly repeated eigenvalues can make any numerical estimate more delicate.
How to use this calculator
- Select a matrix size from 2 × 2 up to 6 × 6.
- Type each matrix entry into the generated input grid.
- Set the displayed decimal precision and a threshold for high sensitivity.
- Press Calculate to show the result panel above the form.
- Review eigenvalues, condition numbers, stability labels, and the two plots.
- Use the CSV button for structured data export and the PDF button for a clean report download.
Why sensitivity matters
Eigenvalue condition numbers measure how strongly a matrix eigenvalue reacts to small perturbations in coefficients. In practical modelling, that sensitivity matters because rounding, measurement noise, and parameter drift all alter matrix entries. When κ(λ) stays near 1, the eigenvalue is usually stable. When κ(λ) grows, tiny input changes may cause noticeably different outputs, classifications, or convergence behaviour.
Interpreting the main metrics
This calculator reports each eigenvalue, its estimated condition number, the reciprocal condition, and a qualitative stability label. Spectral radius summarizes the largest eigenvalue magnitude, while minimum, average, and maximum κ(λ) reveal the spread of sensitivity across the spectrum. A narrow spread suggests consistent behaviour. A wide spread signals that some modes of the matrix are substantially more fragile than others.
Normal versus non normal matrices
Normal matrices, including symmetric real matrices, often produce eigenvalue condition numbers close to 1 because left and right eigenvectors remain well aligned. Non normal matrices behave differently. Even when eigenvalues appear simple and distinct, their associated eigenvectors can become nearly orthogonal, sharply increasing κ(λ). That makes the spectrum look deceptively calm while the underlying system remains highly sensitive to perturbations.
Role of the plots
The complex plane plot helps users see where eigenvalues cluster, whether they are repeated, and whether conjugate pairs appear. The bar chart highlights which eigenvalues dominate sensitivity risk. Together, these plots support faster diagnosis than tables alone for users. A tight cluster with tall bars often indicates separation problems, while evenly sized bars around 1 commonly indicate a more numerically reliable matrix spectrum.
Where the calculator helps
This tool is useful in numerical linear algebra courses, control analysis, vibration studies, iterative solver preparation, and matrix quality review before simulation. Analysts can compare alternative model forms, test perturbation exposure, and flag unstable formulations early. In engineering workflows, that saves time by revealing fragile spectra before downstream optimization, forecasting, or discretization stages depend on unreliable eigenvalue estimates.
Practical reading of results
A strong workflow is to begin with the example matrix, compare it against the identity matrix, then enter your own coefficients and watch how κ(λ) changes. If one eigenvalue shows a very large condition number, inspect scaling, matrix structure, and near defective behaviour. Stable results usually combine moderate magnitudes, separated eigenvalues, and left right eigenvectors that are not almost orthogonal. Repeating the test after small coefficient edits reveals sensitivity patterns more clearly.
FAQs
What does a condition number near 1 mean?
It usually means the eigenvalue is well conditioned. Small matrix perturbations are less likely to cause large first order changes in that eigenvalue.
Why can triangular matrices still look sensitive?
Distinct diagonal entries do not guarantee stability. If the matrix is non normal, left and right eigenvectors may align poorly and increase eigenvalue sensitivity.
Can this calculator handle complex eigenvalues?
Yes. Real input matrices can generate complex conjugate eigenvalue pairs, and the plots place them on the complex plane for visual interpretation.
Why are repeated eigenvalues harder to assess?
Repeated or nearly repeated eigenvalues reduce numerical separation. That makes pairing vectors and estimating sensitivity less robust, especially for defective or nearly defective matrices.
Is a large spectral radius the same as a large condition number?
No. Spectral radius measures magnitude, while the condition number measures sensitivity. An eigenvalue can be large yet well conditioned, or modest in size yet fragile.
When should I export CSV or PDF results?
Use CSV for spreadsheet analysis, documentation, or batch comparison. Use PDF when you need a clean static summary for reporting, teaching, or review meetings.