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Counterexample bank

Part C is won by knowing which tempting claims are false. 148 counterexamples; 85 free. The rest come with the Notes pack.

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#1 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Same characteristic polynomial ⇒ similar” — false

Counterexample: 0 matrix and [[0,1],[0,0]]

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

linear algebracanonical forms

#2 · Analysis & Linear Algebra › Eigenvalues and Canonical Forms

“Same characteristic and minimal polynomial ⇒ similar” — false

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

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

linear algebracanonical forms