Matrix Representation of Adjoints with Respect to Orthonormal Basis

Theorem

1. Let $V\in \mathcal{L}(v)$
2. With $\beta =\{{\alpha }_1,\dots ,{\alpha }_n\}$ as an ordered Orthonormal Basis of $V$
3. With $[T{]}_{\beta }=(A_{jk}$
4. The matrix representation $T^{\ast }$ with basis $\beta$ is the Conjugate Transpose of the matrix representation of $T$ using the same basis

Proof

1. Let $A=[T{]}_{\beta }$
2. Let $B=[T^{\ast }{]}_{\beta }$
3. $⟹A_{jk}=⟨T{\alpha }_k|{\alpha }_j⟩$ and $B_{kj}=⟨T^{\ast }{\alpha }_j|{\alpha }_k⟩$
4. But, $B_{kj}=⟨T^{\ast }{\alpha }_j|{\alpha }_k⟩=\overline{⟨{\alpha }_k|T^{\ast }{\alpha }_j⟩}=\overline{⟨T{\alpha }_k|{\alpha }_j}={\overline{A}}_{jk}$