Is the following statement true? Same characteristic and minimal polynomial ⇒ similar

No. 4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1} Both have χ = x⁴ and m = x², but different numbers of blocks (ranks 2 vs 1).

Counterexample

Is this true?

Same characteristic and minimal polynomial ⇒ similar

No — it is false.

The counterexample

4×4 nilpotent matrices with Jordan blocks of sizes {2, 2} and {2, 1, 1}

Why it fails

Both have $\chi = x^{4}$ and $m = x^{2}$ , but different numbers of blocks (ranks 2 vs 1).

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Execution slip

The idea was right. The computation was not.

Same characteristic polynomial ⇒ similar
Diagonalisable ⇒ invertible
Real matrix with real eigenvalues is diagonalisable

See real exam options built on this mistake Drill statements like this

linear algebra canonical forms

Go deeper· Explained in full in Characteristic vs minimal polynomial — what each tells you· Asked in CSIR NET December 2025· From Eigenvalues and Canonical Forms › Eigenvalues, characteristic & minimal polynomials, Cayley–Hamilton

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

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