For a polynomial f(z) = a0 + a1z + ... +a(n-1)z(n-1) + z(n), a0,..., a
(n-1) is-an-element-of C, with (complex) unimodular zeros, we present
two different methods to construct, by a finite number of rational ope
rations and square roots, a unitary companion matrix, i.e., a unitary
matrix whose characteristic polynomial is (-1)(n)f(z). We also show th
at rational operations alone cannot suffice. Finally, we discuss the p
roblem of constructing a normal companion matrix for a polynomial with
out any restrictions on its zeros and arrive at a negative answer.