Assume:
a sequence of positive semidefinite matrices $A_k^i\in M_{m,n}, m<n$, $i\in\mathcal{N}$, and they have the following property: $$\underline{a}I \leq A_k^i(A_k^i)^T\leq \overline{a}I.$$ In addition, their sum is positive definite and uniformly bounded by $$\underline{\gamma}I\leq \sum_{i\in \mathcal{N}} (A_k^i)^T A_k^i \leq \overline{\gamma} I. $$ Next, we introduce a convex combination to $(A_k^i)^T A_k^i$, $i\in \mathcal{N}$, such that $$\underline{\gamma}'I\leq \sum_{i\in \mathcal{N}} \omega^i (A_k^i)^T A_k^i$$ where $1>\omega^i>0$, and $\sum_{i\in\mathcal{N}}\omega^i =1$.
My question is how to carefully choose $\omega^i$ such that $\sum_{i\in \mathcal{N}} \omega^i (A_k^i)^T A_k^i$ has a best lower bound $\underline{\gamma}^\ast$ ($\underline{\gamma}^\ast\geq\underline{\gamma}'$), in our setting, it is required $\underline{\gamma}^\ast$ to be as bigger as possible.