Let $E_j$ be the $j$th largest-magnitude eigenvalue of a real symmetric $N \times N$ matrix $M$. I've found that the ratio
$$\frac{|E_1|}{\sum_{j=1}^N{|E_j|}},$$
is a measure of the "rank-one-ness" of $M$. Qualitatively, the more similar the columns of $M$ are to each other, the higher the ratio. In my graduate research, this measure appears naturally for a specific class of matrices.
I'm certain that there's been prior research on the properties and usefulness of this measure for deciding how well-aligned and similar the columns of a matrix are. For example, I've seen it used as a measure of "compressibility". Still, my searches haven't turned up much.
Where can I find out more?