Are positive definite matrices robust to "small changes"? Let $A$ be a positive-definite matrix and let $B$ be some other symmetric matrix. Consider the matrix
$$
C=A+\varepsilon B.
$$
for some $\varepsilon>0$. Is it true that for $\varepsilon$ small enough $C$ is also positive definite?

Positive definite means that $\langle v, Av \rangle > 0$ for all nonzero vectors $v$; actually it suffices to check this condition for unit vectors. We have
$$\langle v, Cv \rangle = \langle v, Av \rangle + \epsilon \langle v, Bv \rangle.$$
Now, by the compactness of the unit sphere, $\langle v, Av \rangle$ takes on a minimum nonzero value $m$ on unit vectors (the smallest eigenvalue of $A$, although we don't need this), and $| \langle v, Bv \rangle |$ takes on a maximum nonzero value $M$ on unit vectors (the largest eigenvalue of $B$ in absolute value). 
So we can take $\epsilon < \frac{m}{M}$, which gives
$$\langle v, Cv \rangle \ge m - \epsilon M > 0$$
for all unit vectors $v$. So $C$ is positive definite as desired. 
