Find matrix for total angular momentum along y; find eigenvalues and eigenvectors.

Homo Novus

1. The problem statement, all variables and given/known data

Consider the angular momentum operator [itex]\vec{J_{y}}[/itex] in the subspace for which j=1. Write down the matrix for this operator in the usual basis (where [itex]J^{2}[/itex] and [itex]J_{z}[/itex] are diagonal). Diagonalize the matrix and find the eigenvalues and orthonormal eigenvectors.

2. Relevant equations

[itex]\vec{J} = \vec{L} + \vec{S}[/itex] (total angular momentum)

3. The attempt at a solution

I know J is the sum of angular momentum, L, and spin angular momentum, S, but how to we get it in matrix form? Spin would just be [itex]\hbar / 2[/itex] times the y Pauli matrix... but how do we express L in matrix form? Also, I really don't understand how to obtain eigenvalues and eigenvectors... Could someone go through the problem for me? Textbook is Griffiths. Thanks in advance.