If you are willing to replace $SL$ by $GL$ (and none of the comments/answers so far have mentioned without the determinant) then this is a special case of a much more general result.
Apply the theory in the following paper to the quiver with one vertex and $m$ edges.
MR0958897 (90e:16048) Le Bruyn, Lieven; Procesi, Claudio
Semisimple representations of quivers.
Trans. Amer. Math. Soc. 317 (1990), no. 2, 585–598.
I believe that the
The special case you are considering was studied before this. The reference has already been given by Agol.