Conditional number of synchronization-time matrix

I was reading through this 2011 publication in journal Chaos titled "A phase-synchronization and random-matrix based approach to multichannel time-series analysis with application to epilepsy". Here is the link to the paper: www.ncbi.nlm.nih.gov/pmc/articles/PMC3172996/

I would like to make some observations on the section of "choice of diagonal elements" in this paper:

1. As any synchronization-time matrix (STM for short) is a symmetric matrix, so all its Eigenvalues are real. Further STM is also a normal matrix, which implies that determinant = product of Eigenvalues.
2. Since any STM has all real & positive (>=0) matrix elements, a STM is a non-negative matrix by nature. Hence by the "Perron Frobenius theorem", only the highest Eigenvalue (perron root) is assured to be positive. But a zero diagonal STM is NOT positive definite, so an odd number of its Eigenvalues have to be negative. Further, the smallest Eigenvalue has to be negative.
3. Since adding some constant yet positive number (denoted 'a') to diagonal elements of STM, just results in a constant addition of the same value ('a') to all the Eigenvalues of the zero-diagonal STM. Thus choosing 'a' to be negative of the smallest Eigenvalue of a zero-diagonal STM (which is negative, as noted in observation-[2]) results in a smallest Eigenvalue of modified STM to be zero. Hence the new conditional number C becomes extremely high when 'a' is chosen to be close to the absolute of the smallest Eigenvalue of the zero-diagonal STM.
4. Invoking levy–desplanques theorem, one can say that the choice of diagonal elements for a given STM such that it ensures row dominance (same as column dominance in this case) is a sufficient condition for STM to be positive definite (though this is not a necessary condition).
5. Hence, choosing diagonal elements to be the sum of all non-diagonal elements (denoted 's_nd') is also a sufficient condition. Note: 's_nd' is the same as the average of all non-diagonal elements of STM with an enhancement factor 'alpha' = N, the dimension of STM. However, higher the (chosen) constant value added to diagonal elements, lower is the variance of the condition number and thus lesser is its ability to capture any characteristic changes in the underlying system. 
6. Instead, one can choose 'a' = max{row sum of STM} (it is the same as max{column sum of STM}) without the need to compute the Eigenvalues of STM for each frame of data. This value is much lower than 's_nd' hence a better choice by the same argument.
7. Now that the positive definiteness of STM is assured, the determinant is always positive which enables us to use the same for the analysis.

Overall comments:

• They invoke random matrix theory, but isn't this theory applicable to those random matrices whose non diagonal elements have zero mean and variance = 1/N^2 , where N is the order of the matrix. Since this clearly isn't the case with STM - then why do they choose to invoke this theory?
• The enhancement factor is chosen such that the average of all non-diagonal elements of the STM (denoted as 'avg_nd') dominates the determinant of the STM, following this the determinant of the STM is proposed to be used as a metric. If so then why do they even need this whole analysis? They could as well have chosen to characterize the multi-channel ensemble using just 'avg_nd'. 
• On the other hand, the choice of constant diagonal element to be 'a' = max {row sum of STM} (proposed here in observation-[6]) may give a different characterization than what the proposed 'avg_nd' is currently providing. However this proposal needs to be validated.