We present an eecient algorithm for generating an n n nonsingular matrix uniformly over a nite eld. This algorithm is useful for several cryptographic and checking applications. Over GFF2] our algorithm runs in expected time M(n) + O(n 2), where M(n) is the time needed to multiply two n n matrices, and the expected number of random bits it uses is n 2 + 3. (Over other nite elds we use n 2 + O(1...

In order to have a better understanding of finite random matrices with non-Gaussian entries, we study the 1/N expansion of local eigenvalue statistics in both the bulk and at the hard edge of the spectrum of random matrices. This gives valuable information about the smallest singular value not seen in universality laws. In particular, we show the dependence on the fourth moment (or the kurtosis...

We consider products of independent large random rectangular matrices with independent entries. The limit distribution of the expected empirical distribution of singular values of such products is computed. The distribution function is described by its Stieltjes transform, which satisfies some algebraic equation. In the particular case of square matrices we get a well-known distribution which m...

given four complex matrices a, b, c and d where a 2 cnn and d 2 cmm andlet the matrix(a bc d)be a normal matrix and assume that  is a given complex number that is not eigenvalue of matrix a. we present a method to calculate the distance norm (with respect to 2-norm) from d to the set of matrices x 2 cmm such that,  be a multiple eigenvalue of matrix(a bc x). we also nd the nearest matrix ...

The Circular Law for random matrices with independent log-concave rows Radosław Adamczak (University of Warsaw) I will show how the replacement principle by Tao and Vu together with Klartag’s thin shell inequality and simple bounds for the smallest singular value allow for a relatively easy proof of the circular law for the class of matrices mentioned in the title.

The Restricted Isometry Constants (RIC) of a matrix A measures how close to an isometry is the action of A on vectors with few nonzero entries, measured in the `2 norm. Specifically, the upper and lower RIC of a matrix A of size n ×N is the maximum and the minimum deviation from unity (one) of the largest and smallest, respectively, square of singular values of all `N k ́ matrices formed by taki...

The linear vector–valued channel x 7! QiM ix + z with z and M i denoting additive white Gaussian noise and independent random matrices, respectively, is used to model shadowing for BLAST– like communication systems. It is analyzed in the asymptotic regime as the dimensions of the matrices and vectors involved become large by means of free probability theory. It turns out that almost all singula...