Suppose I have a data set $X \in \mathbb{R}^{n \times d}$, where $n$ is the number of samples and $d$ is the number of measurements. I find that the largest eigenvalue of $X^TX$ is a very large number.
Someone else comes along and says they have done some normalization, and they get a much smaller largest eigenvalue for $X^TX$. In addition, our classifier is doing better on the new data set too.
What can we say about this situation? Is the new data set more separable? Is the spectral norm of $X$ an indicator of some property?