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Part CCSIR NET December 2025a-single-jordan-block-has-only-one-eigenvector-direction-triangularization-still-always-succeeds-though

A single jordan block has only one eigenvector direction triangularization still always succeeds though

Let V be a 4-dimensional complex vector space and A a linear operator on V. Which of the following statements are necessarily true?

  1. A.There exist λ∈C\lambda\in\mathbb{C} and a non-zero v∈V such that Av=λv=\lambda{}v.✓
  2. B.There exist λ,μ∈C\lambda,\mu\in\mathbb{C} and linearly independent vectors v,w∈V such that Av=λv=\lambda{}v and Aw=μw=\mu{}w.
  3. C.There exist λ,μ,δ∈C\lambda,\mu,\delta\in\mathbb{C} and linearly independent vectors v,w∈V such that Av=λv=\lambda{}v and Aw=μv+δw=\mu{}v+\delta{}w.✓
  4. D.There exists a three-dimensional subspace W such that Aw∈W for all w∈W.✓

You have the answer. Trap Analysis is why the other three were written.

Not a worked solution repeated four times — the specific reasoning error each wrong option was built to reward.

See pricing

50 are analysed free — try those first.

Related counterexample: Same characteristic polynomial ⇒ similar

More on this topic

The chapter behind this: Characteristic vs minimal polynomial — what each tells you — free to read

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

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