Hadamard matrices are matrices such $H * H^T = nE$, and all columns are pairwise orthogonal(and all raws are pairwise orthogonal)
The Hadamard matrices are equivalent(~) if it is possible to obtain one from the other by pairwise swapping rows (columns) or multiplying a row (column) by -1.
And the question : is this statement always true $H \sim H^T$?
I have been trying to obtain something good for quite a while. Assuming that $H \sim H^T$, quite a few good properties arise, but I cannot find anything substantial.
For example, if $H \sim H^T$, then $H = Q*H^T*M$, where $Q$ and M are matrix products that interchange two columns (rows) and multiply a column (row) by -1. From this, it follows that $Q^-1 = Q^T$, and similarly for $M$.