We study the spectral norm of matrices W that can be factored as W = BA, where A is a random matrix with independent mean zero entries and B is a fixed matrix. Under the (4 + ε)-th moment assumption on the entries of A, we show that the spectral norm of such an m×n matrix W is bounded by √ m + √ n, which is sharp. In other words, in regard to the spectral norm, products of random and determinis...

Let Fn be an n by n symmetric matrix whose entries are bounded by n γ for some γ > 0. Consider a randomly perturbed matrix Mn = Fn +Xn, where Xn is a random symmetric matrix whose upper diagonal entries xij , 1 ≤ i ≤ j, are iid copies of a random variable ξ. Under a very general assumption on ξ, we show that for any B > 0 there exists A > 0 such that P(σn(Mn) ≤ n−A) ≤ n−B .

This short note is about the singular value distribution of Gaussian random matrices (i.e. Gaussian Ensemble or GE) of size N. We present a new approach for deriving the p.d.f. of the singular values directly from the singular value decomposition (SVD) form, which also takes advantage of the rotational invariance of GE and the Lie algebra of the orthogonal group. Our method is direct and more g...

Let ξ be a real-valued random variable of mean zero and variance 1. Let Mn(ξ) denote the n × n random matrix whose entries are iid copies of ξ and σn(Mn(ξ)) denote the least singular value of Mn(ξ). The quantity σn(Mn(ξ)) 2 is thus the least eigenvalue of the Wishart matrix MnM ∗ n. We show that (under a finite moment assumption) the probability distribution nσn(Mn(ξ)) 2 is universal in the sen...

We apply the method of determinants to study the distribution of the largest singular values of large real rectangular random matrices with independent Cauchy entries. We show that statistical properties of the largest singular values are different from the Tracy-Widom law. Among other corollaries of our method we show an interesting connection between the mathematical expectations of the deter...

In this paper, a numerical method for solving the constrained optimal control of time-varying singular systems with quadratic performance index is presented. Presented method is based on Bernste in polynomials. Operational matrices of integration, differentiation and product are introduced and utilized to reduce the optimal control of time-varying singular problems to the solution of algebraic ...

in this note‎, ‎we obtain some singular values inequalities for positive semidefinite matrices by using block matrix technique‎. ‎our results are similar to some inequalities shown by bhatia and kittaneh in [linear algebra appl‎. ‎308 (2000) 203-211] and [linear algebra appl‎. ‎428 (2008) 2177-2191]‎.

Introducing a new method for studying general probability distributions on Rn, we generalize some results about the least singular value and condition number of random matrices with i.i.d. Gaussian entries to whole class rows.