I'm investigating the eigenvalue ratio

$$\frac{\lambda_1}{\sum_{j=2}^N\lambda_j}$$

of the matrix product $B=AA^T$. $\lambda_1$ denotes the largest eigenvalue. The ratio can be thought of as a measure of "rank-1-ness" of $B$.

I haven't found any mention of this ratio in literature, regardless of constraints on $B$. Has any work been done on this before, and could someone point me there?