1. Jul 9, 2010

### 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)??

2. Jul 9, 2010

### trambolin

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

3. Jul 10, 2010

Ok, thanks!