Proof that transpose of Hadamard Matrix is also a Hadamard matrix The question is self explaining from the title, but let me elaborate it. In most of the articles/books I've read, fact that the transpose of Hadamard matrix is also a Hadamard matrix is used, but I was not able to find or deduce a proof for it. I can basically state that transposing a matrix will (probably?) not affect its orthogonality, but I would prefer a correct proof. 
Thank you,

What you need is the fact, from elementary linear algebra, that, if a square matrix $A$ has a right inverse $B$ (i.e., $AB=I$) then $B$ is also a left inverse for $A$ (i.e., $BA=I$).  In the case of an $n\times n$ Hadamard matrix, the definition in terms of orthogonal rows gives $\frac1n AA^\top=I$. So the linear algebra fact gives $\frac1n A^\top A=I$, which is the characterization by orthogonal columns.
