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Is this true?

Same characteristic polynomial ⇒ similar

No — it is false.

The counterexample

0 matrix and [[0,1],[0,0]]

Both have char poly $x^{2}$ , different minimal polynomials (x vs $x^{2})$ .

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 canonical forms

Others that fail the same way

From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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