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A contractive $n$ -tuple $A\,=\,({{A}_{1}},...,{{A}_{n}})$ has a minimal joint isometric dilation $S\,=\,({{S}_{1}},...,{{S}_{n}})$ where the ${{S}_{i}}$ ’s are isometries with pairwise orthogonal ranges. This determines a representation of the Cuntz-Toeplitz algebra. When $A$ acts on a finite dimensional space, the wot-closed nonself-adjoint algebra $\mathfrak{S}$ generated by $S$ is completely described in terms of the properties of $A$ . This provides complete unitary invariants for the corresponding representations. In addition, we show that the algebra $\mathfrak{S}$ is always hyper-reflexive. In the last section, we describe similarity invariants. In particular, an $n$ -tuple $B$ of $d\,\times \,d$ matrices is similar to an irreducible $n$ -tuple $A$ if and only if a certain finite set of polynomials vanish on $B$ .

Isometric Dilations of Non-Commuting Finite Rank n-Tuples