Is it possible to replace a transformation matrix (square matrix) by a unitary matrix?

Question

I have a square $n\times n$ matrix $A$ which acts on an $n\times1$ vector $x$ to give a particular vector $y$ as a result. Is it possible to produce a unitary matrix which gives the same result $y$ when acting upon $x$?

Answer 1

Community wiki answer so the question can be marked as answered:

As noted in a comment, this is not always possible, as a unitary matrix leaves the norm of a vector invariant, whereas your original matrix might change the norm.