Is the sum of a positive semidefinite matrix and positive definite matrix a positive definite matrix?
I have a positive semidefinite matrix $M\in \mathbb{R}^{n\times n}$ and the identity $I_n$, is their sum positive definite? I ask because I want to ensure their sum has an inverse.
A matrix $M$ is positive-definite (semidefinite) if and only if it is symmetric and $u^TMu>0\space (\geq)$ for all nonzero vectors $u$.
If $M$ is positive definite and $N$ is positive semidefinite then $M$ and $N$ are symettric so their sum $M+N$ is symettric too, and for every $u\neq 0$, $u^TMu>0 $ and $u^TNu\geq 0$, so $u^T(M+N)u>0$, and thus $M+N$ is positive definite.