  We study the structure of isometries defined on the algebra $\mathcal{A}$ of
upper-triangular Toeplitz matrices. Our first result is that a continuous
multiplicative isometry $\mathcal{A}\to M_n$ must be of the form either
$A\mapsto UAU^*$ or $A\mapsto U\overline AU^*$, where $\overline A$ is the
complex conjugation and $U$ is a unitary matrix. In our second result we use a
range of ideas in operator theory and linear algebra to show that every linear
isometry $\mathcal{A}\to M_n(\mathbb{C})$ is of the form $A\mapsto UAV$ where
$U$ and $V$ are two unitary matrices. This implies, in particular, that every
such an isometry is a complete isometry and that a unital linear isometry
$\mathcal{A}\to M_n(\mathbb{C})$ is necessarily an algebra homomorphism.
