I'm investigating the eigenvalue ratios

$$ \frac{\lambda_1}{\sum_{j=2}^N\lambda_j} \quad\mbox{and}\quad \frac{\sum_{j=1}^N\lambda_j}{\sum_{j=2}^N\lambda_j} $$

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

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