Matrix of a restriction of a quadratic form

Question

Given a symmetric matrix $M$ and its associated quadratic form $Q : \mathbb{R}^n\times\mathbb{R}^n \rightarrow \mathbb{R}$ is there an obvious way to write down a matrix of the quadratic form $Q|_{U}$, the restriction of $Q$ to $U$ which is a subspace, for example a hyperplane, of the original space $\mathbb{R}^n$?

Answer 1

If $B$ is a matrix whose columns form a basis for the subspace $U$, then $$ A = B^\top MB $$ is the matrix of the quadratic form relative to this basis. For an example and a detailed explanation of why this is the case, see my post here.