Matrix representation of spin-2 system? I am surprised no one has asked this before, but what is the matrix representation of a spin-2 system?
Also, what are the equivalent of the Pauli matrices for the system?

The irreducible representation of $su(2)$ corresponding to spin 2 is 5-dimensional. One possible choice of explicit $5\times5$ matrices for spin-2 angular momentum is
$$J_1=\left(
\begin{array}{ccccc}
 0 & 1 & 0 & 0 & 0 \\
 1 & 0 & \sqrt{\frac{3}{2}} & 0 & 0 \\
 0 & \sqrt{\frac{3}{2}} & 0 & \sqrt{\frac{3}{2}} & 0 \\
 0 & 0 & \sqrt{\frac{3}{2}} & 0 & 1 \\
 0 & 0 & 0 & 1 & 0 \\
\end{array}
\right)$$
$$J_2=\left(
\begin{array}{ccccc}
 0 & -i & 0 & 0 & 0 \\
 i & 0 & -i \sqrt{\frac{3}{2}} & 0 & 0 \\
 0 & i \sqrt{\frac{3}{2}} & 0 & -i \sqrt{\frac{3}{2}} & 0 \\
 0 & 0 & i \sqrt{\frac{3}{2}} & 0 & -i \\
 0 & 0 & 0 & i & 0 \\
\end{array}
\right)$$
$$J_3=\left(
\begin{array}{ccccc}
 2 & 0 & 0 & 0 & 0 \\
 0 & 1 & 0 & 0 & 0 \\
 0 & 0 & 0 & 0 & 0 \\
 0 & 0 & 0 & -1 & 0 \\
 0 & 0 & 0 & 0 & -2 \\
\end{array}
\right)$$
You can verify that they are Hermitian; that the eigenvalues of each one are -2, -1, 0, 1, and 2; that
$$[J_i,J_j]=i\epsilon_{ijk}J_k;$$
and that
$$J_1^2+J_2^2+J_3^2=2(2+1)I.$$
This paper discusses the construction of spin matrices for arbitrary spin.
