The following paper describes all invariants for the diagonal conjugation action on $m$-tuples of matrices. This is not exactly what you want (because you second action is not conjugation), but you can try to adapt the methods to describe a full set of invariants.

• MR0419491 (54 #7512)
  Procesi, C. The invariant theory of n×n matrices. Advances in Math. 19 (1976), no. 3, 306–381.

• MR0419490 (54 #7511) Reviewed Procesi, Claudio The invariants of n×n matrices. Bull. Amer. Math. Soc. 82 (1976), no. 6, 891–892.

From the review: The classical groups $GL(n,\mathbb C)$, and their maximal compact subgroups act by conjugation on m-tuples of $n\times n$ complex matrices $(X_1,\dots,X_m)$. The author announces without proof results about the corresponding invariants. $\dots$

The following paper describes all invariants for the diagonal conjugation action on $m$-tuples of matrices. This is not exactly what you want (because you second action is not conjugation), but you can try to adapt the methods to describe a full set of invariants.

• MR0419491 (54 #7512)
  Procesi, C. The invariant theory of n×n matrices. Advances in Math. 19 (1976), no. 3, 306–381.

• MR0419490 (54 #7511) Reviewed Procesi, Claudio The invariants of n×n matrices. Bull. Amer. Math. Soc. 82 (1976), no. 6, 891–892.

From the review: The classical groups $GL(n,\mathbb C)$, and their maximal compact subgroups act by conjugation on m-tuples of $n\times n$ complex matrices $(X_1,\dots,X_m)$. The author announces without proof results about the corresponding invariants. $\dots$