I want to see the difference between just adjoint and self-adjoint (hermitian) operator represented by matrices.
If I have a matrix $$A= \begin{pmatrix} 1 & i \\ i & 1 \\ \end{pmatrix}$$ then its conjugate is
$$A^*= \begin{pmatrix} 1 & -i \\ -i & 1 \\ \end{pmatrix}$$ ?
Or what are the properties of just adjoint matrices?
The Hermitian matrix is for example: $$A= \begin{pmatrix} 1 & -i \\ i & 1 \\ \end{pmatrix}$$
because its conjugate transpose equals to $A$.
I am very confused as I saw just the definitions of adjoint operators via scalar product, like this: $[Ax,y] = [x,A^*y]$ but I cannot think about an example to get intuition even though my question probably is pretty obvious.