Suppose we have $M\geq 3$ numbers of symmetric matrices $\mathbf{A}_1, \mathbf{A}_2, \cdots, \mathbf{A}_M \in \mathbb{R}^{n\times n}$. Each of them has $n$ numbers of real eigenvalues:
$$ \lambda_1^1\leq \lambda^1_2 \cdots\leq \lambda^1_n\\ \lambda_1^2\leq \lambda^2_2 \cdots\leq \lambda^2_n\\ \vdots\\ \lambda_1^M\leq \lambda^M_2 \cdots\leq \lambda^M_n, $$ fortunately, we also have $\lambda^1_1>0$ and $\lambda^M_1>0$ ($\mathbf{A}_1$ and $\mathbf{A}_M$ are positive definite).
Then, we are interested in the eigenvalues of $$ \mathbf{S} = \mathbf{A}_1 + \mathbf{A}_2+ \cdots+ \mathbf{A}_M. $$
I have questions:
If I want $\lambda_\text{min}(\mathbf{S})$ to be positive, are there any sufficient and necessary conditions on the eigenvalues of $\mathbf{A}_1, \mathbf{A}_2, \cdots, \mathbf{A}_M$? (i.e. Find the lower bound of $\lambda_\text{min}(\mathbf{S})$)
more soft question: are there any useful theorems for the analysis of this?