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Part CCSIR NET December 2023reading-jordan-blocks

Reading jordan blocks

Suppose a 7 × 7 block diagonal complex matrix A has blocks (0), (1), [[0, 1], [0, 0]] and the 3×3 block with 2πi2\pi{}i on the diagonal, a 1 in position (1,2) and zeros elsewhere, along the diagonal. Which of the following statements are true?

  1. A.The characteristic polynomial of A is x3(x−1)(x−2πi)3x^{3}(x - 1)(x - 2\pi{}i)^{3}.✓ Correct
  2. B.The minimal polynomial of A is x2(x−1)(x−2πi)3x^{2}(x - 1)(x - 2\pi{}i)^{3}.
  3. C.The dimensions of the eigenspaces for 0,1,2πi0, 1, 2\pi{}i are 2, 1, 3 respectively.
  4. D.The dimensions of the eigenspaces for 0,1,2πi0, 1, 2\pi{}i are 2, 1, 2 respectively.✓ Correct

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, and why this one tests invariants don't determine.

See pricing

50 are analysed free — try those first.

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

The chapter behind this: Jordan canonical form — free to read

From Eigenvalues and Canonical Forms › Jordan canonical form

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