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Is this true?

Every real matrix has a Jordan form over

No — it is false.

The counterexample

The rotation [[0,−1],[1,0]]

Its eigenvalues ±i are not real; over one uses the real Jordan form with a 2×2 rotation block.

The kind of mistake this is

Invariants don't determine

Two objects sharing an invariant were treated as the same object.

Drill statements like this
linear algebra

Others that fail the same way

From Eigenvalues and Canonical Forms › Jordan canonical form

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