Complex endomorphism/Complex and real determinant/Exercise

Let ${}V$ be a finite-dimensional vector space over the complex numbers ${}\mathbb {C}$, and let

$\varphi \colon V\longrightarrow V$

be a ${}\mathbb {C}$-linear mapping. We consider ${}V$ also as a real vector space of double dimension. ${}\varphi$ is also a real-linear mapping, which we denote by ${}\psi$. Show that between the complex determinant and the real determinant, the relation

${}\vert {\det \varphi }\vert ^{2}=\det \psi \,$

holds.

Create a solution