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Largest-to-rest Has the largest-to-rest eigenvalue ratio of real symmetric matrices been researched before?

I'm investigating the eigenvalue ratioratios

$$\frac{\lambda_1}{\sum_{j=2}^N\lambda_j}$$$ \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 product $B=AA^T$. $\lambda_1$ denotes the largest eigenvalue. The ratio ratios can be thought of as a measure of "rank-1-ness" of $B$.

I haven't found any mention of this either ratio in literature, regardless of constraints on $B$. Has any work been done on this before, and could someone point me there?
show/hide this revision's text 1

Largest-to-rest eigenvalue ratio of real symmetric matrices

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?