Is $Q + A$ postive semidefinte? Let $Q$ be a Matrix and $V \subseteq \Bbb R^n$ be a vector subspace on which  $Q$ is positive semidefinite i.e,  $$\langle x  ,  Q x \rangle \ge 0 ~~~~~  \quad \forall x \in V $$
Prove or provide a counter example
There exist a matrix $A$ such that $Q + A$ is positive semidefinite (on whole $\Bbb R^n$) and 
$$\langle x  ,  Q x \rangle  = \langle x  ,  (Q + A) x \rangle   \quad \quad x \in V.$$

If $P$ is a projection onto $V$, then for any $x \in \mathbb{R}^n$ we have
$$\langle x, P^T QPx \rangle = \langle Px, QPx \rangle \ge 0$$
since $Px \in V$, so $P^T QP$ is positive semidefinite. Moreover if $x \in V$ then $Px = x$, so
$$\langle x, P^T QPx \rangle = \langle Px, QPx \rangle = \langle x, Qx \rangle.$$
So pick $A = P^T QP - Q$ and we're done.
