Determinants whose elements have equal norm
Theodore S. Motzkin · Proceedings of the American Mathematical Society · 1960
Hence A, B, C equal A', B', C' in some order. Each of the 6 orders leads immediately to the proportionality of two rows or columns. The above theorem, in its specialization to minors of Vandermonde determinants composed of qth roots of unity in R*, was used in [1] for the proof of a theorem on power series without termns whose subscript belongs to one of 3 residue classes modulo an arbitrary integer q. In [2 ] I showed that the corresponding theorem for 4 residue classes is false for q = 6. This suggests that There exist vanishing 4 by 4 minors of the form J ,ajbk J,,q= 1,-, e in R*, without proportional rows or columns. Indeed, examining the counterexample in [2] in the light of the proof in [1] we obtain the determinant