all. I want to ask your opinion about the following question:
Let $\mathbf{A} \in \mathbb{R}^{m \times n}$ be a matrix having full column-rank, and let $\{ \mathbf{u}_{1}, \cdots, \mathbf{u}_{r} \}$ be the set of vectors in $\mathbb{R}^{n}$ such that $\{ \mathbf{A} \mathbf{u}_{1}, \cdots, \mathbf{A} \mathbf{u}_{r} \}$ is orthonormal. Then, does the following inequality hold true? Here, $\mathbf{A}^{T}$ denotes the transpose of $\mathbf{A}$.
\begin{align} \underset{i=1}{\overset{r}{\sum}} \| \mathbf{A}^{T} \mathbf{A} \mathbf{u}_{i} \|_{2}^{2} &\ge \frac{r^{2}}{\| \underset{i=1}{\overset{r}{\sum}} \mathbf{u}_{i} \|_{2}^{2}} \end{align}
When $r=1$, then the above inequality clearly holds true by the well-known Cauchy-Schwarz inequality ($\| \mathbf{A}^{T} \mathbf{A} \mathbf{u}_{1} \|_{2} \cdot \| \mathbf{u}_{1} \|_{2} \ge \langle \mathbf{A}^{T} \mathbf{A} \mathbf{u}_{1}, \mathbf{u}_{1} \rangle = \| \mathbf{A} \mathbf{u}_{1} \|_{2}^{2} = 1$).
How about the case for $r \neq 1$?