Maximum eigenvalue of $XX^T$

Question

Can we say something about the maximum eigenvalue of $XX^T$ given that $||x_i||_2 \leq 1$ for all $i$?

Is the largest eigenvalue of $XX^T$ also $\leq 1$? Is it $\leq n$? Or is there some bound?

I'm just asking because this problem I'm solving has this condition $||x_i||_2 \leq 1$ that I haven't used in my solution, and I don't know how to use it.

Answer 1

Note that $\|X X^T\| = \|X^T\|^2$ and $X^T u = \begin{bmatrix} e_1^T X^T u \\ \vdots \\ e_n^T X^T u \end{bmatrix}$ so $\|X^T u\| = \sqrt{\sum_k |e_1^T X^T u|^2} \le \sqrt{\sum_k \|e_1^T X^T \|^2}\|u\|= \sqrt{n} \|u\|$.

Hence the maximum eigenvalue of $X X^T$ is bounded by $n$.

To get equality, choose $X= { 1 \over \sqrt{n}} e e^T$, where $e=(1,...,1)^T$.