Prove that matrices made entirely of blocks of the form $\begin{pmatrix} z & iw \\ i \bar w & \bar z \end{pmatrix}$ have a real determinant.

For example, we claim $$\Delta=\det \begin{pmatrix} z_1 & iw_1 &z_2 & i \bar w_1\\ i \bar w_1 &\bar z_1 & i\bar w_2 & \bar z_2\\ z_3 & iw_3 &z_4 & i \bar w_4\\ i \bar w_3 &\bar z_3 & i\bar w_4 & \bar z_4 \end{pmatrix} \in \mathbb{R}\tag{*}$$ for all $z_i,w_i \in \mathbb{C}$.

Why this is interesting

If we start with an arbitrary square matrix $A$ over the quaternions $\mathbb{H}$, and send each element of the form $a+bi+cj+dk$ to the matrix $\begin{pmatrix} a+bi & c+di \\ -c+di & a-bi \end{pmatrix}$, we can define the determinant of $A$ as the determinant of the matrix obtained after expanding $A$ to a complex matrix in that way.

Turns out this is the "right" choice, in conserving important rules like invertibility being equivalent to nonzero determinant. So in other words, we need to:

Prove all quaternionic matrices have a real determinant.

For a single block the claim is obvious, but it is not clear how to reduce the general case to it.


To see that $\Delta\in\mathbb{R}$ let us take the complex conjugate of it and show that $\overline{\Delta}=\Delta$ in (*). It will be clear what happens with all the blocks by looking at the first block only. $$ \overline{\Delta}=\det\overline{\begin{pmatrix} z & iw \\ i \bar w & \bar z \end{pmatrix}}= \det\begin{pmatrix} \bar z & -i\bar w \\ -iw & z \end{pmatrix}. $$ Now we switch every odd row (in total $n$ many) with the next even row and then every odd column (also $n$ many) with the next even one. The determinant changes the sign $2n$ times, i.e. keeps the same one. $$ \det\begin{pmatrix} \bar z & -i\bar w \\ -iw & z \end{pmatrix}=-\det\begin{pmatrix} -iw & z\\ \bar z & -i\bar w \end{pmatrix}=\det\begin{pmatrix} z & -iw \\ -i\bar w & \bar z \end{pmatrix}. $$ Now multiply every odd column and then every odd row by $-1$. The determinant is multiplied by $(-1)^{2n}=1$ $$ \det\begin{pmatrix} z & -iw \\ -i\bar w & \bar z \end{pmatrix}=-\det\begin{pmatrix} -z & -iw \\ i\bar w & \bar z \end{pmatrix}=\det\begin{pmatrix} z & iw \\ i\bar w & \bar z \end{pmatrix}=\Delta. $$ Since $\overline{\Delta}=\Delta$ it is real.


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