How do I find adjoint operator in different bases?

Question

I know the Gram matrix $G$ of euclidean space my operator is in and the matrix $A$ of it. How do I use them to calculate adjoined operator? I know I can use $A^*=\overline{G^{-1}A^TG}$ So for example if

$G= \begin{pmatrix} 1 & 1 & 1 & 1\\\ 1 & 2 & 2 & 2 \\\ 1 & 2 & 3 & 3 \\\ 1 & 2 & 3 & 4\end{pmatrix}$

and

$A= \begin{pmatrix} 0 & 1 & 0 & 0\\\ 1 & 0 & 1 & 0 \\\ 1 & -1 & 0 & 1 \\\ 0& 0 & 0 & 1\end{pmatrix}$

i get

$G^{-1}A^TG = \begin{pmatrix} 0 & 1 & 0 & 0\\\ 2 & 3 & 4 & 3 \\\ -1 & -3 & -3 & -2 \\\ 0& 1 &1 & 1\end{pmatrix}$

but how do I get $\overline{G^{-1}A^TG}$?

Answer 1

Since the matrix ${G^{-1}A^TG}$ has real entries, its conjugate is the same of the matrix:

$\overline{G^{-1}A^TG}={G^{-1}A^TG}$