Transpose of $Y ={A}^{*} \operatorname{diag} \left( b \right) {A} $

Question

What would the tranpose of $Y$ look like $$Y ={A}^{*} \operatorname{diag} \left( b \right) {A} $$

where $*$ is the conjugate tranpose, i.e., hermitian.

Answer 1

We have $\operatorname{diag}(b)^*=\operatorname{diag}(\overline{b})$, hence

$Y^*=A^*\operatorname{diag}(\overline{b})A^{**}=A^*\operatorname{diag}(\overline{b})A$.