Could somebody give me an example of a matrix that doesn’t admit a Jordan Canonical Form over $\mathbb R$ and explain why it does not?
2026-03-29 18:32:27.1774809147
What is a matrix with no Jordan canonical form?
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A matrix admits a Jordan canonical form just when all its eigenvalues lie in the base field. (See https://en.wikipedia.org/wiki/Jordan_normal_form).
So all you need is a matrix with at least one nonreal eigenvalue. Write down the matrix for a quarter circle rotation in the plane.