I am trying to show that a regular Hadamard matrix must have order $m^2$ for some integer $m$.
So far I have found that if $H$ is an $r$-regular Hadamard matrix of order $n$, then $HJ = rJ$ and $HH^TJ = nJ$.
Does that mean if I can prove that $H=H^T$, then it follows that $n = r^2$ ?
A regular Hadamard matrix has all row and column sums equal, for instance $$\pmatrix{1&-1&1&1\\1&1&-1&1\\1&1&1&-1\\-1&1&1&1}.$$
ADDED IN EDIT
In a regular Hadamard matrix, $Hu=ru$ and $H^Tu=ru$ where $u$ is the all-ones column vector and $r$ is the row/column sum. Therefore $$HH^Tu=H(ru)=r^2u.$$ But $HH^T=nI$ by Hadamard-ness, so $$HH^Tu=nu.$$ Therfore $n=r^2$.