What is the relationship between det(M) and det(X+iY)?

[quasar987]

Let X and Y be n x n real matrices such that the 2n x 2n matrix M:=

X -Y
Y X

is invertible. What is the relationship between det(M) and det(X+iY)??

 

[trambolin]

With a congruence transformation
<br /> \begin{pmatrix}<br /> I &amp;-iI\\-iI &amp;I<br /> \end{pmatrix}<br /> \begin{pmatrix}<br /> X &amp;-Y\\<br /> Y &amp;X<br /> \end{pmatrix}<br /> \begin{pmatrix}<br /> I &amp;iI\\iI &amp;I<br /> \end{pmatrix}<br /> = 2\begin{pmatrix}<br /> X-iY &amp;0\\0 &amp;X+iY<br /> \end{pmatrix}<br />
Hence the determinant is equal to the product determinant.

 

[quasar987]

Ok, thanks!