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Part CCSIR NET June 2023kernel-dimension-vs-block-sizes

Kernel dimension vs block sizes

Let V be a finite dimensional real vector space and be two nilpotent operators on V. Let {} and {}. Which of the following statements are FALSE?

  1. A.If and are similar, then and are isomorphic vector spaces.
  2. B.If and are isomorphic vector spaces, then and have the same minimal polynomial.✓
  3. C.If , then and are similar.
  4. D.If and are isomorphic, then and have the same characteristic polynomial.

Solution

dim W = number of Jordan blocks. Equal numbers of blocks do not fix the largest block (minimal polynomial): blocks {2,2} vs {3,1} in dimension 4. (1) similar ⇒ equal kernel dimension. (3) W = V ⇒ T = 0. (4) nilpotent ⇒ characteristic polynomial always.

The trap it tests

Invariants don't determine

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

Drill statements like this

Related counterexample: Every real matrix has a Jordan form over ℝ

More on this topic

From Eigenvalues and Canonical Forms › Jordan canonical form

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