Eigenvalue Sensitivity Calculator - Matrix Perturbation Analysis

Matrix Input Form

Use the responsive calculator grid below. Large screens show three columns, smaller screens show two, and phones show one.

Matrix Size

Finite Difference Step

Display Decimals

a11

a12

a13

a21

a22

a23

a31

a32

a33

Scenario Entry Row

Scenario Entry Column

Scenario Perturbation Value

Example Data Table

Matrix Size	Matrix Entries	Finite Difference Step	Scenario Entry	Scenario Delta	Purpose
3 x 3	[[5, 1, 0], [1, 4, 1], [0, 1, 3]]	0.0001	a23	0.05	Check which entry changes the spectrum most strongly.
2 x 2	[[4, 1], [2, 3]]	0.0001	a11	0.02	Inspect local eigenvalue sensitivity near a baseline system.

Formula Used

For a matrix A, the calculator estimates eigenvalues numerically with a shifted QR iteration. It then perturbs one entry at a time and measures how each eigenvalue moves.

The first order sensitivity idea is:

∂λ_k / ∂a_ij ≈ [λ_k(A + hE_ij) - λ_k(A)] / h

Here, h is the finite difference step and E_ij is a matrix with 1 in position i,j and 0 elsewhere. The calculator reports absolute sensitivity magnitudes and a scaled relative sensitivity:

Relative sensitivity ≈ |∂λ_k/∂a_ij| × |a_ij| / max(|λ_k|, ε)

This gives a practical stability clue for small perturbations. It is most useful when the matrix is not defective and the chosen step is reasonably scaled.

How to Use This Calculator

Select a 2 x 2 or 3 x 3 matrix.
Enter all matrix coefficients in the input grid.
Choose a finite difference step for derivative estimates.
Pick the row, column, and perturbation value for the scenario test.
Set the number of decimals you want displayed.
Press Calculate Sensitivity to show results above the form.
Review eigenvalue shifts, entry sensitivities, and the Plotly charts.
Download the output as CSV or PDF when needed.

Frequently Asked Questions

1. What does eigenvalue sensitivity measure?

It measures how strongly an eigenvalue reacts when a matrix entry changes slightly. Larger values mean that small coefficient changes can produce larger spectral movement.

2. Why does the calculator use a finite difference step?

Finite differences approximate derivatives numerically. They are practical when you want entry-by-entry sensitivity without deriving symbolic formulas for every matrix case.

3. What step size should I choose?

Start with a small value such as 0.0001. If results look noisy, try a slightly larger step. If the step is too large, local derivative accuracy may weaken.

4. Can the calculator handle complex eigenvalues?

Yes. When the final 2 x 2 block produces a negative discriminant, the result is shown as a complex pair using real and imaginary parts.

5. What does relative sensitivity tell me?

Relative sensitivity scales the derivative by entry size and eigenvalue size. It helps compare importance across entries when raw values alone are hard to judge.

6. Why might two nearby eigenvalues look unstable?

Closely spaced eigenvalues can exchange order or react sharply to tiny perturbations. That often signals a more delicate spectral structure and higher practical sensitivity.

7. Is this tool suitable for defective matrices?

Use extra caution. Defective or nearly defective matrices can show very large sensitivity, and derivative estimates may depend strongly on scaling and perturbation choice.

8. What do the charts show?

The bar chart compares absolute sensitivity across matrix entries for each eigenvalue. The second chart compares baseline and scenario eigenvalue magnitudes after the chosen perturbation.