Letbe a partition of the positive integer α. The block diagonal subgroupof the general linear groupGlα(K) (Kis a field of characteristic 0) acts on then-tuples of α × α matrices overKby simultaneous conjugation. We give the first and second fundamental theorems for the polynomial invariants of this action. We translate the second fundamental theorem into the language of block trace identities, and show that all of them are consequences of the fundamental trace identities. The results give generators and relations for the polynomial invariants of representations of quivers