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In my recent studies of quaternions, the following orthogonal matrix has come up. For example, it is related to the matrix representation of quaternion multiplication. Has anyone seen it come up in other studies? $$A = \frac{1}{2}\begin{bmatrix}1 & -1 & -1 & -1\\ 1 & 1 & -1 & 1 \\ 1 & 1 & 1 & -1\\ 1 & -1 & 1 & 1 \end{bmatrix}$$

In general I'm interested in $4^n \times 4^n$ orthogonal matrices where all the entries are $\pm \frac{1}{2^n}$. So if anyone has a link with information about these kinds of orthogonal matrices (or how they are related to each other) it would be very much appreciated. Or block matrices that are orthogonal and the non-zero blocks are of this form.

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As already pointed out by comments, your matrix is related to Hadamard matrices. You can use Sylvester's construction to build up these matrices by using $$ \begin{bmatrix} H & H\\ H & -H\end{bmatrix} , $$ but there are also other constructions. A nice list of matrices, including dimensions other than the one you requested, can be found here.

Further, Hadamard matrices can be complex too, see here.

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