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?



Has the largest-to-rest eigenvalue ratio of real symmetric matrices been researched before?
I am not sure if you are allowed to change your objective function, but a natural alternative for measuring "rank-1-ness" is 
$$
\frac{\lambda\_1^2}{\lambda\_1^2+\lambda\_2^2+\dots+\lambda\_n^2}.
$$
This ratio is easy to compute: the denominator is the squared Frobenius norm of the matrix (i.e., sum of squares of all entries), and the numerator is the squared spectral radius, which you can easily estimate using a few iterations of the power method.


The norm $\lambda\_1+\lambda\_2+\dots+\lambda\_n$ is called *nuclear norm*, and as far as I know it is not as easy to compute as the other two.


