The purpose of this paper is to locate and estimate the left eigenvalues of quaternionic matrices. We present some distribution theorems for the left eigenvalues of square quaternionic matrices based on the generalized Gerschgorin theorem and generalized Brauer theorem.

Ostrowski type and Brauer type theorems are derived for the left eigenvalues of quaternionic matrix. We see that the above theorems for the left eigenvalues are also true for the case of right eigenvalues, when the diagonals of quaternionic matrix are real. Some distribution theorems are given in terms of ovals of Cassini that are sharper than the Ostrowski type theorems, respectively, for the ...

Many real world networks are reported to have hierarchically modular organization. However, there exists no algorithm-independent metric to characterize hierarchical modularity in a complex system. The main results of the paper are a set of methods to address this problem. First, classical results from random matrix theory are used to derive the spectrum of a typical stochastic block model hier...

We consider a class of matrices of the form Cn = (1/N)(Rn+σXn)(Rn+σXn) ∗, where Xn is an n × N matrix consisting of independent standardized complex entries, Rj is an n×N nonrandom matrix, and σ > 0. Among several applications, Cn can be viewed as a sample correlation matrix, where information is contained in (1/N)RnR ∗ n, but each column of Rn is contaminated by noise. As n → ∞, if n/N → c > 0...