Positive definiteness of a linear combination of semi positive definite matrices?

Question

Assume I have the following condition: $M=\sum_{i=1}^N M_i$ is a positive definite matrix while $M_i$ is semi-positive definite matrix. If we introduce positive integer $\alpha_i>0$ such that $M'=\sum_{i=1}^N \alpha_i M_i$, my question is that does $M'$ has the same positive definiteness as $M$, or simply, does the $M'$ and $M$ have the same rank?

Answer 1

Positive definiteness of $M$ implies positive definiteness of $M'$, as $$ M' = \alpha M + \sum_{i=1}^N \underbrace{(\alpha_i - \alpha)}_{\ge 0} M_i $$ for $$\alpha = \min_{1\le i \le N} \alpha_i > 0.$$

The converse is not true.