Is it true that $\lambda_{\min} \left(\sum_{i=1}^m \mathbf{u}_i\mathbf{u}_i^t \right) \le \max_{1=1}^m \|\mathbf{u}_i\|_2^2$, for $m>n$?

Question

Let the vectors $\{\mathbf{u}_i\}_{i=1}^m\in \mathbb{R}^n,\ m>n$ be given. Does the following hold?

$$\lambda_{\min} \left(\sum_{i=1}^m \mathbf{u}_i\mathbf{u}_i^t \right) \le \max_{1=1}^m \|\mathbf{u}_i\|_2^2$$

If we had $m\le n$, I could have just argued that $$\lambda_{\min}(\sum_{i=1}^m \mathbf{u}_i\mathbf{u}_i^t)\le \frac{tr(\sum_{i=1}^m \mathbf{u}_i\mathbf{u}_i^t)}{n}\le \frac{m}{n}\max_i\|\mathbf{u}_i\|_2^2\le \max_i\|\mathbf{u}_i\|_2^2.$$ But this argument does not work for $m>n$. Does this still hold for $m>n$, or is there any simple counter-example or simple proof of this fact that might have missed my mind?.

Thanks in advance.

Answer 1

Let ${\bf u}_i$ consist of $k$ repetitions of ${\bf e}_1, \ldots, {\bf e}_n$ (the standard unit vectors), so $m = kn$. $\sum_{i} {\bf u}_i {\bf u}_i^t = k I$, so the left side is $k$, but the right side is $1$.