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

 

[quasar987]

Ok, thanks!