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An $n\times n$ matrix $A$ is called a Hadamard matrix if all of its entries are $\pm 1$ and $AA^T=nI$. Show that the Kronecker product $A\otimes B$ of two $n \times n$ Hadamard matrices $A$ and $B$ is again a Hadamard matrix.

This is an interesting result, but I don't know how to start with. It seems that I need to use induction on $n$? Any good idea?

J. W. Tanner
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breezeintopl
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1 Answers1

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It's easy to see that the entries of $A \otimes B$ are $\pm 1$, since each one is the product of an entry of $A$ and an entry of $B$. Also, $(A \otimes B)^T = A^T \otimes B^T$ and so $$ (A \otimes B) (A \otimes B)^T = (A \otimes B) (A^T \otimes B^T) = (AA^T) \otimes (BB^T) = (nI) \otimes (nI) = n^2 I.$$

Yuval Filmus
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