Is there anything significant we can say about matrices of the form $$ \left( \begin{array}{cc} a & b\\ \overline{b} & \overline{a} \end{array} \right) $$ Such a matrix looks almost but not quite like an embedding of the quaternions. Relevance: I'm looking at a matrix partitioned into $2 \times 2$ blocks where every off diagonal block is of the above form and the diagonal blocks are multiples of the identity.
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What do you want to achieve with the big block matrix? Without knowing that it is hard to guess what information about the blocks might be interesting. For instance, the determinant is $|a|^2-|b|^2$, which might be useful, but also the block is a $2×2$ matrix, probably not that useful. – P. Siehr Jul 26 '17 at 11:35
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If you look at all matrices of this form such that $a$ is real, you have a commuting family of self-adjoint matrices, which makes life easier. – Ben Grossmann Jul 26 '17 at 12:28
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One of the form of this kind of block matrix is Vandermonde matrix over $GF(2^q)$. – Amin235 Jul 26 '17 at 12:31