A property regarding the eigenvalue of the sum of positive semidefinite matrices is positive definite?

Question

If the sum of positive semidefinite matrices $A_i$ is positive definite (they are also symmetric), that is $$A=\sum_{i\in \mathcal{N}} A^i, \;\mathcal{N}=\{1,2,\cdots,N\}$$ is positive definite.

What's the relation between the minimal eigenvalue of $A$ and the minimal positive eigenvalue of $A^i$.

Answer 1

Let $\lambda=\inf(spectrum(A))$ and $\lambda_i=\inf(spectrum(A_i))$; then $\inf_{||x||=1}x^TAx=\lambda$ and, for every $i$, $inf_{||x||=1}x^TA_ix=\lambda_i$. Since $x^TAx=\sum_ix^TA_ix\geq \sum_i\lambda_i$, we obtain $\lambda\geq \sum_i \lambda_i$.

EDIT. The previous reasoning is valid for every symmetric matrices $A,A_i$. Moreover we may have: for every $i$, $\lambda_i=0$ and $\lambda>0$.