Is the following statement true? Every real matrix has a Jordan form over ℝ

No. The rotation [[0,−1],[1,0]] Its eigenvalues ±i are not real; over ℝ one uses the real Jordan form with a 2×2 rotation block.

Counterexample

Is this true?

Every real matrix has a Jordan form over $\mathbb{R}$

No — it is false.

The counterexample

The rotation [[0,−1],[1,0]]

Why it fails

Its eigenvalues ±i are not real; over $\mathbb{R}$ one uses the real Jordan form with a 2×2 rotation block.

Keep practicing

Base field or ring

The answer changes with the field or ring you are working over.

Published 23 Aug 2026. Found a mistake? Tell us — corrections are the fastest thing we act on.