Let $M$ be an $n\times n$ matrix of real entries, which has at least one positive eigenvalue. Now consider $f$ a function of class $C ^{2}$ in a domain of $\mathbb{R} ^ n$, and $g$ a continuous function defined in the same domain. It is given that $f (0) = g (0) = 0$, and $g <f$ in a ball centered on the origin and radius $r> 0$.
Aff: $g-h$ reaches minimum at a point $x_{0}$ in $B_{r}$ for $\epsilon>0$ small enough.
Where $h (x) = f (x) + e | P_{S} x |$,
Here $P_{S} x$ is the orthogonal projection on $S$, and $S$ is the direct sum of the self-spaces associated with positive eigenvalues.
I have no idea how the orthogonal projection comes in handy here, so I don't know how to start proving that statement. A tip, or a sketch of the race, helps me a lot.
PS: If it helps you can exchange $f$ for a quadratic polynomial.