Let $n$ be a positive integer. A matrix $A = [a_{ij}] \in \mathcal{M}_{n\times n}(\mathbb{C})$ is a complex Hadamard matrix if and only if $|a_{hj}| = 1$ for all $1 \leq h$, $j \leq n$ and any pair of distinct rows of $A$, considered as vectors in $\mathbb{C}_n$, is orthogonal. For each $n$, find a complex number $d$ such that $A = [d^{hj}]$ is a complex Hadamard matrix. For $n = 6$, find a complex Hadamard matrix which is not of this form.

I have no idea where to begin on this guy. Any help is greatly appreciated. Thanks in advance


I know this is very old, but in case someone else comes by:

For the first part (finding the $d$), you probably want a [rational] root of unity, right? After all, you're taking lots of powers of it, and its absolute value is 1. If you're still stuck, try thinking about Fourier transform matrices.

For the second part -- finding a different one for $n=6$ -- start by finding two different solutions for $n=3$ and putting them side-by-side. This will give you 3 orthogonal rows. Then you can try extending that out to add more rows.


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