Spin operator matrix representations in Sx basis
The solution is quite simple, and based on 2 observations -
- You expect the $S_x$ to be diagonal (with appropriate eigenvalues)
- The commutation relations of spin operators must be preserved (in some books the spin operators are defined by those relations) ($[S_i,S_j]=i\hbar \varepsilon_{ijk}S_k$)
The solution is to make a permutation of the known operators, i.e $S_x \rightarrow S_z$, $S_z \rightarrow S_y$, $S_y \rightarrow S_x$.
$$S_x = {\hbar \over 2}\begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$$ $$S_z = {\hbar \over 2}\begin{pmatrix} 0 & -i \\ i & 0\end{pmatrix}$$ $$S_y = {\hbar \over 2}\begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}$$ The $S^2$ operator, being rotationaly invariant (proportional to unit matrix), remains unchanged $$S^2 = {3 \hbar^2 \over 4}\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$$
Another way to solve this is to look at this as purely algebraic problem - find the unitary matrix which diagonalizes $S_x$ and apply it to all the remaining matrices (changing representation basis of the matrix).