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$A$ is a Hadamard matrix of side $n$, and $H$ is a Hadamard matrix of side $m+n$, where $H=\pmatrix{A& B \\\ C& D}$ for some matrices $B,C,D$. Prove that $m \geq n$.

A Hadamard matrix is a square matrix of side n with entries -1,1 and $HH^t=nI$. I'm assuming I have to use the following facts that I know about Hadamard matrices: The rows and columns are pairwise orthogonal.

Every HM can be normalized ($1$'s in the first row and column)

$n+m$ is 1, 2 or a multiple of $4$

I'm not sure how to prove please.

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up vote 2 down vote accepted

Try writing out the product $HH^t$ in block form. One of the blocks will be $AA^t+BC^t$. What must this block be equal to if $H$ is Hadamard? What can you say about the rank of $B$ if $m<n$? Can you produce a contradiction in this way?

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