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Part CCSIR NET December 2025rank-0-means-only-the-zero-matrix-so-that-orthogonality-condition-is-vacuous-not-restrictive

Rank 0 means only the zero matrix so that orthogonality condition is vacuous not restrictive

For n≥3, consider the space Mn(C)M_{n}(\mathbb{C}) of n×n complex matrices endowed with the inner product ⟨A,B⟩ = Trace(A*B). For 0≤k≤n, let Wk=W_{k} = {A∈Mn(C)A \in M_{n}(\mathbb{C}) : ⟨A,B⟩=0 for all B∈Mn(C)B \in M_{n}(\mathbb{C}) with rank k}. Which of the following statements are necessarily true?

  1. A.W0=W_{0} = {0}
  2. B.W1=W_{1} = {0}✓ Correct
  3. C.W2=W_{2} = {0}✓ Correct
  4. D.Wn=W_{n} = {0}✓ 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 boundary and endpoint.

See pricing

50 are analysed free — try those first.

The trap it tests

Boundary and endpoint

The statement turns at the edge of the interval, the domain, or the parameter range.

Drill statements like this

Related counterexample: A real matrix with all real eigenvalues is orthogonally diagonalisable

More on this topic

The chapter behind this: Inner products, orthogonality and the spectral theorem — free to read

From Inner Product Spaces and Forms › Gram–Schmidt, orthogonal/unitary/normal matrices, spectral theorem

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