Unitary Eigenvalues Calculator

Matrix settings

Matrix size (n×n)

Larger sizes may take more time to solve.

Unit-circle tolerance

Used for the “On unit circle?” check.

Decimal places

Enter matrix entries

Allowed: 2, -3.1, i, -i, 2+3i, 2-0.5i.

Actions

What you get

Eigenvalues as complex numbers.
Magnitudes and angles for each eigenvalue.
Unitarity error using ||U·Uᴴ − I||.
CSV and PDF downloads for the results table.

Example data table

Matrix size	Example entry pattern	Expected eigenvalue magnitudes
2×2	Diagonal phases: e^{iθ1}, e^{iθ2}	All should be 1
3×3	Normalized DFT-like matrix	All should be 1
4×4	Rotations with phase shifts	All should be 1

Formula used

A complex matrix U is unitary when U·Uᴴ = I, where Uᴴ is the conjugate transpose.

Eigenvalues λ satisfy det(U − λI) = 0. For unitary matrices, each eigenvalue lies on the unit circle, so |λ| = 1.

This tool builds the characteristic polynomial using Faddeev–LeVerrier, then finds complex roots using the Durand–Kerner iteration.

How to use this calculator

Select the matrix size and set a tolerance.
Enter each element using real or complex form.
Click “Compute Eigenvalues” to show results above.
Review |λ| and the unit-circle status.
Download CSV or PDF for sharing.

FAQs

1) What is a unitary matrix?

A matrix is unitary when its conjugate transpose equals its inverse. That identity preserves lengths and angles of vectors under multiplication.

2) Why do unitary eigenvalues have magnitude one?

If Uv=λv, then ||Uv||=||v|| for unitary U. This gives |λ|·||v||=||v||, so |λ|=1 for any nonzero eigenvector.

3) Why is |λ| not exactly 1 sometimes?

Floating-point rounding and iterative root solving can cause tiny drift. Non-unitary inputs also push magnitudes away from 1. Increase decimals or check the unitarity error.

4) What does the unitarity error measure?

It is ||U·Uᴴ − I|| using the Frobenius norm. A value near zero means U is close to unitary within numeric precision.

5) What complex formats are accepted?

Use 2, -3.1, i, -i, 2+3i, or 2-0.5i. Keep “i” at the end and avoid spaces for best parsing.

6) What sizes work best?

2×2 to 6×6 are supported. Smaller sizes converge faster. For 6×6, choose reasonable tolerances and decimals to avoid slow iteration.

7) Is this suitable for symbolic math?

No. Results are numeric approximations. For exact symbolic eigenvalues, use a computer algebra system and compare numeric values with this tool.