Trades in complex Hadamard matrices
A trade in a complex Hadamard matrix is a set of entries which can be
changed to obtain a different complex Hadamard matrix. We show that in
a real Hadamard matrix of order n all trades contain at least n
entries. We call a trade rectangular if it consists of a
submatrix that can be multiplied by some scalar c ≠ 1 to obtain
another complex Hadamard matrix. We give a characterisation of
rectangular trades in complex Hadamard matrices of order n and show
that they all contain at least n entries. We conjecture that all
trades in complex Hadamard matrices contain at least n entries.