Suppose that $c_i \in \mathbb{R}-\{0\}, B_i \in \mathbb{R}^{k \times m}, \alpha \in \mathbb{R}^k$ with $\|\alpha\|_2 = 1$. Consider the following linear combination of outer products:
$$M = \sum_{i=1}^n c_i B_i^\intercal \alpha \alpha^\intercal B_i$$
I'm looking for a lower bound on the following ratio
$$\frac{\text{tr}(M)}{\|M\|_2}$$
Specifically, I want to examine if this ratio is lower bounded by an increasing function of dimension $m$ (such as $\sqrt{m}$ or $\log m$) based on the values of $c_i$ and properties of matrices $B_i$ such as norm, trace, eigenvalues, etc.
My attempt: If $M$ is positive semi-definite, then above ratio is lower bounded by $1$ since
$$\frac{\text{tr}(M)}{\|M\|_2} = \frac{\lambda_1+\dots +\lambda_m}{\lambda_1} \geq 1,$$
where $\lambda_1 \geq \dots \geq \lambda_n$ are eigenvalues of $M$. Equality is achieved when $\lambda_1 = 1, \lambda_{i>1} = 0$. This ratio becomes an increasing function of $m$ when eigenvalues of $M$ do not decay very fast. Also, trace of $M$ is given by
$$\text{tr}(M) = c_i \alpha^\intercal (\sum_{i=1}^n B_i B_i^\intercal) \alpha $$
Is there a way to relate eigenvalues of $M$ to properties of $B_i$? Any reference on this topic is appreciated.