Calculator Inputs
Enter a 2x2 matrix, choose a normalization style, and compare the spectral radius with a risk threshold.
Formula Used
Characteristic equation: det(A − λI) = 0
Eigenvalues for a 2x2 matrix: λ = (trace(A) ± √(trace(A)2 − 4det(A))) / 2
Eigenvector rule: Solve (A − λI)v = 0 for any nonzero direction vector v.
Risk lens: The spectral radius ρ(A) = max(|λ|) measures whether shocks decay, persist, or amplify.
In risk management, eigenvectors reveal dominant directions of exposure movement, while the spectral radius shows whether feedback loops are shrinking or intensifying.
How to Use This Calculator
- Enter the four values of your 2x2 matrix.
- Name the scenario, such as transition risk, credit migration, or loss propagation.
- Choose whether eigenvectors should stay raw, be unit-normalized, or scale to a first component of 1.
- Set the stability threshold. A common value is 1 for propagation analysis.
- Press Calculate Eigenvectors to display results above the form.
- Use the CSV or PDF buttons to export the summary and compare scenarios later.
Example Risk Matrix
| Scenario | Matrix A | Trace | Determinant | Approx. λ1 | Approx. λ2 | Comment |
|---|---|---|---|---|---|---|
| Market Shock Dampening | [[0.82, 0.15], [0.10, 0.91]] | 1.73 | 0.7312 | 0.9955 | 0.7345 | Largest mode stays near 1, so persistence remains high. |
| Escalating Contagion | [[1.15, 0.28], [0.21, 1.08]] | 2.23 | 1.1832 | 1.4022 | 0.8278 | One dominant mode exceeds 1 and amplifies exposure. |
Frequently Asked Questions
1. What does this calculator solve?
It finds eigenvalues, eigenvectors, determinant, trace, discriminant, condition number, and spectral radius for any 2x2 matrix. It also interprets the dominant mode from a risk propagation viewpoint.
2. Why are eigenvectors useful in risk management?
Eigenvectors show the directions where exposures move together. They help identify stable portfolios, persistent transition paths, and the combinations of factors most responsible for amplification.
3. What does the spectral radius tell me?
The spectral radius is the largest absolute eigenvalue. Below your threshold, shocks tend to decay. At the threshold, behavior is marginal. Above it, feedback can grow.
4. What happens if both eigenvalues are the same?
A repeated eigenvalue can still have one or many valid eigenvectors. Scalar matrices allow every nonzero vector. Other repeated cases may collapse into a single eigendirection.
5. Can this calculator handle complex eigenvalues?
Yes. When the discriminant is negative, it shows complex conjugate eigenvalues and their matching complex eigenvectors. The graph switches to the complex plane automatically.
6. Which normalization option should I choose?
Use unit normalization to compare directions fairly, raw vectors for algebra checks, and first-component scaling when you want a standardized ratio-based representation.
7. Why is the condition number missing sometimes?
The condition number needs an inverse matrix. If the determinant is zero, the matrix is singular, so inversion fails and the condition number is not defined.
8. Is this the same as singular value analysis?
No. Eigen-analysis studies invariant directions under the matrix itself. Singular values measure stretching strength, even when eigenvectors are not the best stability description.