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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}✓
  3. C.W2=W_{2} = {0}✓
  4. D.Wn=W_{n} = {0}✓

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.

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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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