In a book whose title I don’t remember I read the following problem from S.W. Drury. $U^*$ denotes the conjugate transpose of $U$. Show that every $5 \times 5$ matrix $U$ with complex entries $u_{j,k}$ of constant absolute value one that satisfies $U^*U=5I$ can be realized as the matrix $(\omega^{j,k})_{j,k}$ where $\omega$ is a complex primitive fifth root of unity by applying some sequence of the following. (1) A rearrangement of the rows. (2) A rearrangement of the columns. (3) Multiplication of a row by a complex number of absolute value one. (4) Multiplication of a column by a complex number of absolute value one.
I didn’t find any solution but I would like to know if we could just find a matrix that satisfies those conditions. And is there a procedure to verify numerically those conditions ? Many thanks.