Operator Eigenvalue Calculator

Formula used

For a linear operator represented by a square matrix A, eigenvalues λ satisfy the eigenvalue equation A v = λ v for a non‑zero vector v. Equivalently, they are the roots of the characteristic equation:

det(A − λI) = 0

This calculator estimates eigenvalues using a shifted QR iteration, which repeatedly factorizes A − μI = Q R and updates A ← R Q + μI. When the off‑diagonal terms shrink, the diagonal approaches the eigenvalues.

How to use this calculator

1. Select the matrix size n×n for your operator.
2. Enter the matrix entries in the grid, row by row.
3. Set a tolerance and maximum iterations for convergence control.
4. Enable eigenvectors if you also need approximate eigenmodes.
5. Press Calculate to see results above the form.
6. Use CSV or PDF buttons to export the computed report.

Example data table

Operator eigenvalues in physics

1) Eigenvalues and measurement outcomes

Observables are modeled by linear operators, and eigenvalues are the values obtained in ideal measurements. Energy levels, spin projections, and normal‑mode frequencies are all spectra of operators.

2) Operator to matrix representation

Choosing a basis turns an operator into a square matrix A. A change of basis applies A → S⁻¹AS, which preserves eigenvalues, so the spectrum is basis‑independent.

3) Characteristic equation

Eigenvalues λ satisfy det(A − λI) = 0. For larger matrices, numerical estimation is preferred over expanding high‑degree polynomials, especially when data are noisy, ill‑conditioned, or strongly scaled.

4) Hermitian and symmetric cases

Hermitian (real‑symmetric) operators have real eigenvalues and orthogonal eigenvectors, matching physical requirements for measured quantities. If your input is nearly symmetric, small asymmetries can shift results.

5) Degeneracy and close eigenvalues

Degeneracy occurs when multiple eigenvectors share one eigenvalue, often due to symmetry. Near‑degeneracy appears as clustered values; using tighter tolerances and more iterations helps separate them.

6) Data examples you can test

A = [[1, 0.2],[0.2, 2]] gives λ ≈ 0.9615 and 2.0385 (level splitting). The tensor [[5,1,0],[1,4,0],[0,0,3]] gives ≈ 5.6180, 3.3820, 3.0000 (principal values). The Laplacian [[2,-1,0],[-1,2,-1],[0,-1,2]] gives ≈ 3.4142, 2.0000, 0.5858 (mode structure).

7) Numerical method used

This tool uses shifted QR iteration: factorize A − μI = QR, then update A ← RQ + μI. As off‑diagonal terms shrink, the diagonal approaches the eigenvalues. Practical controls such as maximum iterations, tolerance, and an optional shift improve convergence for tough matrices.

8) Interpreting and validating results

Eigenvalues carry the operator’s units (energy, s⁻¹, etc.). As quick checks, compare trace(A) to the sum of eigenvalues and det(A) to their product (small n), and confirm expected symmetry. For stability operators, eigenvalues with positive real parts often indicate growth or instability. If results look unexpected, rescale inputs and rerun.

FAQs

1) What does this calculator compute?

It estimates the eigenvalues of a linear operator represented by your input square matrix. Eigenvalues satisfy A v = λ v and are returned as a numerical list, with optional real/imaginary parts when needed.

2) Which matrices work best?

Real‑symmetric matrices usually converge fastest and return real eigenvalues. General real matrices also work, but may converge slower or produce complex pairs. Well‑scaled inputs improve numerical stability.

3) How large can the matrix be?

You can choose size n and enter an n×n matrix. Very large n increases computation time in the browser and may reduce accuracy. For best results, start with n ≤ 10 and increase gradually.

4) Why do I see small imaginary parts for a symmetric operator?

Tiny imaginary parts typically come from rounding, tolerance settings, or slight asymmetry in inputs. Recheck entries, enforce symmetry (Aᵢⱼ = Aⱼᵢ), tighten tolerance, and rerun.

5) What do tolerance and iterations control?

Tolerance sets how small off‑diagonal terms must become before stopping. Max iterations limits how many QR steps are performed. Lower tolerance and higher iterations generally improve accuracy but take longer.

6) How can I sanity‑check the output?

Compare trace(A) with the sum of eigenvalues, and for small matrices compare det(A) with the product of eigenvalues. If your operator is symmetric, eigenvalues should be real and eigenvectors orthogonal.

7) Can I use this for quantum Hamiltonians?

Yes, if you provide a matrix representation of the Hamiltonian in a chosen basis. For Hermitian Hamiltonians, eigenvalues represent energy levels. Ensure consistent physical units and consider rescaling to avoid extreme magnitudes.