CONSTRUCTION OF UNITARY AND NORMAL COMPANION MATRICES

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.