Determinant of a complex block representation
= Determinant of a complex block representation
{title2=$\det M=\det(A+iB)\det(A-iB)$}
The block matrix $M=\begin{pmatrix}A&-B\\B&A\end{pmatrix}$ is similar over $\mathbb C$ to $\operatorname{diag}(A+iB,A-iB)$ via $T=\begin{pmatrix}I&I\\-iI&iI\end{pmatrix}$. Therefore its <determinant> is $\det(A+iB)\det(A-iB)$ for arbitrary square matrices $A,B$. The argument uses a constant <change-of-basis matrix> and <multiplicativity of the determinant>.