Realification of a complex matrix ($\mathcal{R}(A)=\left(\begin{array}{cc}\mathrm{Re}A& -\mathrm{Im}A\\ \mathrm{Im}A& \mathrm{Re}A\end{array}\right)$)

For $A=P+iQ$ with real matrices $P,Q$, the same linear map viewed over the reals has block matrix $\left(\begin{array}{cc}P& -Q\\ Q& P\end{array}\right)$ in grouped real/imaginary coordinates. Interleaved coordinates merely permute the basis. Its complexification splits into the matrices $A$ and $\overline{A}$, giving the realification determinant identity.

The realification of a complex matrix has real determinant $|{\det }_{\mathbb{C}}A{|}^2$. Indeed the invertible complex coordinate change $(x,y)\mapsto (x+iy,x-iy)$ conjugates its complexified block matrix to $\mathrm{diag}(A,\overline{A})$. Taking determinants gives the identity, including singular matrices.