On matrices with pairwise orthogonal rows and columns

We discuss possible forms of square matrices whose rows are pairwise orthogonal and the same is true of their columns. This discussion is applied to the problem of conditions under which a nonsingular binormal matrix is unitoid. A square matrix $A$ is said to be binormal if the matrices $AA^*$ and $A^*A$ commute. A square matrix is said to be unitoid if it can be brought to diagonal form by a (Hermitian) congruence.