What is the transpose of a conjugate transpose? Let's say we have a matrix $A$ and its conjugate transpose $A^{*}$, would the transpose of the conjugate transpose be the conjugate of $A$, i.e. $\bar{A}$?
Would one represent that as follows:
$A^{*T} = \bar{A}$
I have seen this written out as numpy code, but I don't know how to represent it.
The transpose of the transpose is the original matrix, as transposition is an involution.
So you're right, the transpose of the conjugate transpose of a matrix $A$ is just the conjugate $\bar A$, although I think this notation is not heavily used in linear algebra.