ExampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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The adjoint of an explicit map $C^{2} \to C^{3}$ is its conjugate-transpose matrix

Example

For the map $T : C^{2} \to C^{3}$ with standard matrix

$A = (\begin{array}{cc} 1 & i \\ 2 - i & 0 \\ - i & 3 \end{array}),$

the adjoint $T^{*} : C^{3} \to C^{2}$ has matrix

$A^{*} = (\begin{array}{ccc} 1 & 2 + i & i \\ - i & 0 & 3 \end{array}) .$

Facts & Assumptions

Given: The displayed complex matrix $A$ in standard orthonormal bases.

[L1]

In orthonormal bases, the matrix of the adjoint is the conjugate transpose (In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix).

[L2]

The standard complex coordinate pairing is $⟨ x, y ⟩ = \sum_{k} x_{k} \overline{y_{k}}$ (The standard formulas $⟨ x, y ⟩ = \sum_{k < n} x_{k} y_{k}$ on $R^{n}$ and $\sum_{k < n} x_{k} \overline{y_{k}}$ on $C^{n}$ are inner products).

Verification

technique · computation

1.1

Transpose $A$ and conjugate each entry. This gives the displayed $2 \times 3$ matrix $A^{*}$ , exactly as [L1] prescribes.

L1algebra

2.1

For generic $x \in C^{2}$ and $y \in C^{3}$ , expansion with [L2] gives $⟨ A x, y ⟩ = x^{T} A^{T} \overline{y} = ⟨ x, A^{*} y ⟩$ . Thus the displayed matrix also verifies the adjoint identity directly.

step 1.1L2algebra∎

Depends on

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The adjoint of an explicit map C2→C3C2→C3 is its conjugate-transpose matrix