Examples of non-unitary isometries on finite dimensional Hilbert spaces? I was reading the question A Finite Dimensional non-Unitary Isometry?, which gives an example of a non unitary isometry which is a map $T: R \rightarrow R^2 $. This question is based on a previous question Difference between an isometric operator and a unitary operator on a Hilbert space, in which there is an example of non-unitary isometry in an infinite-dimensional Hilbert space. 
Are there any examples of operators on finite dimensional Hilbert spaces $V: H_A \rightarrow H_A$ which have $V^\dagger V = \mathbb{I}$ but $V V^\dagger \neq \mathbb{I}$, or does isometry imply unitarity in this special case?

For all linear operators $A,B$ on a finite dimensional vectorspace $AB=I$ implies $BA=I$. See here.
