Let $Q$ be a Matrix and $V \subseteq \Bbb R^n$ be a vector subspace on which $Q$ is positive definite i.e, $$\langle x , Q x \rangle > 0 ~~~~~ \quad \forall x \in V \setminus \{0\} $$
Prove or provide a counter example
There exist a matrix $A$ such that $Q + A$ is positive definite (on whole $\Bbb R^n$) and
$$\langle x , Q x \rangle = \langle x , (Q + A) x \rangle \quad \quad x \in V.$$
Note that: This question has been solved for the case Positive semidefinite. Here Is $Q + A$ postive semidefinte?