How to calculate $J_z$ and $exp(itJ_z)$ for irreducible representation of $su(2)$ with $j=\frac{5}{2}$, expressing it as $n\times n$ matrices?
I know that $J_z|j,m\rangle=m|j,m\rangle$, and $m=-j, -j+1,...,j$, so I tried writing $J_z$ for every $m$ like that:
$J_z|\frac{5}{2},-\frac{5}{2}\rangle=-\frac{5}{2}(0\ 0\ 0\ 0\ 0\ 1)^T$
I finally got matrix
\begin{bmatrix} 0&0&0&0&0&\frac{5}{2}\\ 0&0&0&0&\frac{3}{2}&0\\ 0&0&0&\frac{1}{2}&0&0\\ 0&0&-\frac{1}{2}&0&0&0\\ 0&-\frac{3}{2}&0&0&0&0\\ -\frac{5}{2}&0&0&0&0&0 \end{bmatrix}
Is it correct?