Complex Eigenvalue Calculator

Enter Matrix

Type real and imaginary parts. Use i-form outputs and export options after calculation.

Example Data Table

A simple 2×2 rotation matrix produces purely imaginary eigenvalues.

Matrix (A)	Expected eigenvalues	Notes
[[0, -1], [1, 0]]	+0 + 1i, +0 - 1i	Classic 90° rotation; complex conjugate pair.
[[2+1i, 0], [0, 3-2i]]	2+1i, 3-2i	Diagonal matrix; eigenvalues are the diagonal entries.

Formula Used

Eigenvalues are solutions of the characteristic equation det(A − λI) = 0.

2×2 closed form

For A = [[a, b], [c, d]]: tr = a + d, det = ad − bc, λ = (tr ± √(tr² − 4det))/2.

General n×n numerical method

The calculator uses QR iterations: A₀ = A, then Aₖ = RₖQₖ where Aₖ = QₖRₖ. The diagonal of Aₖ approaches eigenvalues.

Note: Unshifted QR is robust for small matrices, but some inputs need more iterations for best accuracy.

How to Use This Calculator

Select a matrix size (2×2 to 5×5).
Enter each cell’s real and imaginary values.
Choose iterations and display precision.
Click Compute Eigenvalues to see results above.
Use CSV/PDF buttons to export the result table.

Input Scope and Matrix Sizes

This tool accepts complex entries a+bi for matrices from 2×2 through 5×5, covering 4 to 25 cells. Each cell is stored as two numeric fields, enabling purely real cases by setting imaginary parts to 0. A diagonal 5×5 instantly returns five diagonal eigenvalues, giving a quick correctness check. For teaching, start with 2×2 and compare against the exact closed form output shown below.

Iteration Control and Convergence Signals

Eigenvalues for general sizes are approximated by QR iterations, with a selectable range of 20 to 400 steps. In practice, many well scaled 3×3 problems stabilize near 60–120 iterations, while tougher non normal matrices may need 200+. A trace consistency check compares trace(A) to Σλ; smaller differences indicate improved convergence. If the trace gap stops shrinking, reduce scaling extremes.

Complex Plane Interpretation

The Plotly chart places eigenvalues at (Re, Im) so conjugate pairs appear mirrored across the real axis. Marker size scales with |λ|, helping you spot dominant modes quickly. For the example rotation matrix [[0,−1],[1,0]], the points land near (0,±1), matching the expected purely imaginary spectrum. Clusters near the origin indicate slow dynamics or oscillations.

Precision, Rounding, and Export Integrity

Display precision is adjustable from 2 to 12 decimals. CSV and PDF exports use the same precision, keeping the numeric trail consistent across reports. If you increase precision but see unstable last digits, raise iterations or rescale your matrix so typical magnitudes sit near 1 to 10.

Performance Notes for 5×5 Cases

A 5×5 run performs repeated QR decompositions, so total work grows quickly with iteration count. For most browser and shared hosting setups, 80 to 160 iterations offers a good balance between speed and accuracy. If your input has very large values, normalize the matrix first to reduce rounding drift. Sparse patterns may converge faster than dense random matrices.

Practical Use Cases and Validation

Complex eigenvalues appear in stability analysis, vibration with damping, control systems, and rotations. Validate results by checking invariants: trace(A) ≈ Σλ and, for 2×2, the closed form λ=(tr±√(tr²−4det))/2. When these align within your precision target, the computed spectrum is usually reliable for downstream modeling. Use the graph to confirm expected symmetry when coefficients are real, and document results for audits easily.

FAQs

1) Why do eigenvalues become complex?

Complex eigenvalues occur when the characteristic polynomial has nonreal roots, often from rotations, oscillations, or non-symmetric coupling. Real matrices typically produce conjugate pairs.

2) Is the 2×2 result exact?

Yes. For 2×2 inputs, the page shows the closed-form solution using trace and determinant. The QR result is also shown for comparison.

3) How many iterations should I use?

Start with 80. If the trace difference remains large or digits fluctuate, raise to 160 or 240. For easy matrices, 40–80 is usually enough.

4) What does the plot tell me?

It plots each eigenvalue at (Re, Im). Distance from the origin equals magnitude |λ|, and symmetry about the real axis indicates conjugate pairing for real-valued matrices.

5) Why does my CSV show rounded values?

Exports follow the selected display precision. Increase precision to capture more decimals, or increase iterations if the last digits are not stable.

6) Can I use this for repeated experiments?

Yes. Keep your matrix entries consistent, export CSV/PDF for each run, and track changes in magnitude and angle to compare scenarios across iterations.