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Part BCSIR NET December 2023schur-invariant-flag

Schur invariant flag

Let A be an n × n matrix with complex entries. If n ≥ 4, which one of the following statements is true?

  1. A.A does not have any non-zero invariant subspace in .
  2. B.A has an invariant subspace in of dimension n − 3.✓
  3. C.All eigenvalues of A are real.
  4. D. does not have any invariant subspace in of dimension n − 1.

Solution

Over every matrix is triangularisable (Schur), so the span of the first k basis vectors of a triangularising basis is an invariant subspace of every dimension k — in particular n − 3 ≥ 1.

The trap it tests

Execution slip

The idea was right. The computation was not.

Drill statements like this

Related counterexample: AB and BA have the same minimal polynomial

More on this topic

From Vector Spaces and Linear Maps › Linear transformations, matrix representation, change of basis

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