I am studying Lie groups and Lie algebras.One of the standard examples of Lie groups is $O(n;k)$,which is called the generalized orthogonal group.It is defined by $O(n;k)=\{T\in GL_n(\mathbb C): [Tx,Ty]_{n,k}=[x,y]_{n,k}\}$ where $[.,.]$ is the bilinear form $[x,y]_{n,k}=\sum\limits_{i=1}^n x_iy_i-\sum\limits_{i=1}^k x_{n+i}y_{n+i}$.I am trying to show the following:
$O(n;k)=\{A\in GL_{n+k}(\mathbb R):A^tgA=g\}$ where $g$ is the diagonal matrix with $1$ in the first $n$ diagonal entries and $-1$ in the last $k$ diagonal entries.
But I am unable to show this.I am sure that this equality is in the isomorphism sense.Can someone tell me a way how to proceed?
Is there any typo anywhere in the above?If yes,please correct.