As everyone know that we can use a matrix $A$ to represent an operator $T$.
The adjoint of a matrix $A$ is denoted as $A^*$, which takes complex conjugate of $A$ and then transpose.
My problem is, what is the relationship between $A^*$ and $A^{-1}$?
I mean as we know $Ax = b$, $x = A^{-1}b$. (If not emphasize on the dimension of $x$ and $b$.
We have $A^{-1}A = I$, hence $A^* (A^{-1})^* = I$, from which it follows that $(A^*)^{-1} = (A^{-1})^*$.