Calculator Inputs
Example Data Table
| Matrix A | Test Value λ | Illustrative σmin(A − λI) | Interpretation |
|---|---|---|---|
| diag(2, 5) | 2 | 0 | Exact spectral point for the matrix. |
| diag(2, 5) | 4 | 1 | Shift is not singular, so it fails the test. |
| [3 1; 0 3] | 3 | 0 | Defective eigenvalue, still part of the spectrum. |
| diag(3, 5) | 2.98 | 0.02 | Near a spectral value under a loose tolerance. |
Formula Used
For a bounded operator, λ belongs to the approximate point spectrum when there exists a sequence of unit vectors xn such that ‖(A − λI)xn‖ → 0.
For a finite matrix, the calculator evaluates the smallest singular value of the shifted matrix A − λI. A small value means the shift is nearly singular and therefore numerically close to the approximate point spectrum test.
Core computation: σmin(A − λI) = min‖x‖=1 ‖(A − λI)x‖.
Decision rule: if σmin(A − λI) ≤ ε, then λ is accepted as an ε-approximate point spectrum candidate.
Relative residual: residual / (‖A‖F + |λ|) helps compare matrices on different scales.
How to Use This Calculator
- Choose the matrix dimension and enter a square real matrix.
- Enter the real and imaginary parts of the test value λ.
- Set a tolerance that reflects how strict the approximation should be.
- Pick the number of displayed decimals for clean output.
- Press Calculate to place the result directly above this form.
- Review σmin, the residual norm, and the interpretation badge.
- Inspect the approximate eigenvector to understand the minimizing direction.
- Use the CSV or PDF buttons to export the computed report.
Frequently Asked Questions
1. What does approximate point spectrum mean?
It describes values of λ for which A − λI can send some unit vectors to very small residuals. In finite dimensions, this matches the usual spectrum, but the residual viewpoint is still valuable numerically.
2. Why does the calculator use singular values?
The smallest singular value gives the minimum possible residual over all unit vectors. That makes it a direct computational test for whether the shifted matrix is nearly singular.
3. Can I enter a complex matrix?
This page accepts real matrix entries, while λ may be complex. That already covers many practical spectral checks for real systems with complex candidate values.
4. What tolerance should I choose?
Use a smaller tolerance for stricter tests and cleaner data. Use a larger tolerance when entries are noisy, scaled strongly, or when you want a broader numerical screening.
5. Does a small residual guarantee an exact eigenvalue?
For finite matrices, very small residuals indicate that A − λI is close to singular. Within numerical tolerance, that often signals an eigenvalue or a point extremely close to one.
6. What is the approximate eigenvector shown in results?
It is the normalized vector that minimizes the residual in the singular-value problem. This vector shows the direction in which the shifted matrix acts almost singularly.
7. Why is the relative residual useful?
Absolute residuals can look large for large matrices or scales. Relative residuals normalize the result, making comparisons easier across different matrices and candidate values.
8. When should I export CSV or PDF?
Export the report when you need to document spectral tests, compare candidate values, or attach numerical evidence to coursework, notes, or technical reviews.