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
$B$.