TITLE: Groups of matrices that preserve several quadratic forms
QUESTION [9 upvotes]: Given two (or more) quadratic forms (on the same vector space) consider the group of matrices that preserve these forms, i.e. $Q_i=U Q_i U^T$, $i=1,2..,k$ What is known about such groups? (at least for k=2 and the forms are symmetric and one is of full rank) 
The keywords? Where to read?

REPLY [2 votes]: A starting point is that such a $U$ commuttes with $Q_2Q_1^{-1}$. This reduces the analysis to the restriction of $U$ to generalized eigenspaces of $Q_2Q_1^{-1}$.
In particular, suppose that $Q_1$ (or some linear combination of the $Q_j$'s) is positive definite. Then one may choose a basis in which $Q_1=I_n$ and $Q_2$ is diagonal. Then $U$ is block diagonal, with each block being associated with one eigenvalue of $Q_2$. In general, the group is very small.