Complex eigenvalues

[tired meadow]

Find a 2×2 matrix A with the following properties: Each eigenvalue is a
complex number with modulus equal to 1. The eigenvalues remain complex
for An for 0 ≤ n ≤ 49 and become real for A50. How to solve this ?

[west stag]

1. Wait patiently for a helper to come along.
2. Once someone helps you, say thank you and close the thread with:

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3. Feel free to nominate the person for helper of the week in #helper-nominations

[oblique spire]

If you allow complex entries, then this question is trivial. If you want A to be a real 2 by 2 matrix, then consider a rotation.
$$\begin{pmatrix}\cos(\pi/50) & -\sin(\pi/50)\ \sin(\pi/50) & \cos(\pi/50)\end{pmatrix}$$

[terse lynx]

Ἵππαρχος

[oblique spire]

The point is that the eigenvalues of a rotation by $\theta$ are $\exp(i\theta)$ and its conjugate, which are unit modulus, and you can easily modify that by using roots of unity in order to satisfy the second criterion.

[terse lynx]

Ἵππαρχος

[oblique spire]

@tired meadow