We study the asymptotic behavior of outliers in the spectrum of bounded rank perturbations of large random matrices. In particular, we consider perturbations of elliptic random matrices which generalize both Wigner random matrices and non-Hermitian random matrices with iid entries. As a consequence, we recover the results of Capitaine, Donati-Martin, and Féral for perturbed Wigner matrices as w...

We present results from using Random Indexing for Latent Semantic Analysis to handle Singular Value Decomposition tractability issues. We compare Latent Semantic Analysis, Random Indexing and Latent Semantic Analysis on Random Indexing reduced matrices. In this study we use a corpus comprising 1003 documents from the MEDLINE-corpus. Our results show that Latent Semantic Analysis on Random Index...

These lectures cover three loosely related topics in random matrix theory. First we discuss the techniques used to bound the least singular value of (nonHermitian) random matrices, focusing particularly on the matrices with jointly independent entries. We then use these bounds to obtain the circular law for the spectrum of matrices with iid entries of finite variance. Finally, we discuss the Li...

We present an approach that allows one to bound the largest and smallest singular values of an N × n random matrix with iid rows, distributed according to a measure on R that is supported in a relatively small ball and linear functionals are uniformly bounded in Lp for some p > 8, in a quantitative (non-asymptotic) fashion. Among the outcomes of this approach are optimal estimates of 1±c √ n/N ...

the linear system of equations ax = b where a = [aij ] in cn.n is a crispsingular matrix and the right-hand side is a fuzzy vector is called a singularfuzzy linear system of equations. in this paper, solving singular fuzzy linearsystems of equations using generalized inverses such as drazin inverse andpseudo-inverse are investigated.

Following the results of cite{Med}, regarding the Aluthge transform of polynomial matrices, the symbolic computation of the Duggal transform of a polynomial matrix $A$ is developed in this paper, using the polar decomposition and the singular value decomposition of $A$. Thereat, the polynomial singular value decomposition method is utilized, which is an iterative algorithm with numerical charac...

We show that, for an n×n random matrix A with independent uniformly anticoncentrated entries such that E‖A‖HS2≤Kn2, the smallest singular value σn(A) of satisfies P{σn(A)≤ε n}≤Cε+2e−cn,ε≥0. This extends earlier results (Adv. Math. 218 (2008) 600–633; Israel J. 227 (2018) 507–544) by removing assumption mean zero and identical distribution across as well recent result (Livshyts (2018)) where was...