The Expected Norm of a Sum of Independent Random Matrices: an Elementary Approach
نویسندگان
چکیده
ABSTRACT. In contemporary applied and computationalmathematics, a frequent challenge is to bound the expectation of the spectral norm of a sum of independent randommatrices. This quantity is controlled by the norm of the expected square of the randommatrix and the expectation of the maximum squared norm achieved by one of the summands; there is also aweak dependence on the dimension of the randommatrix. The purpose of this paper is to give a complete, elementary proof of this important, but underappreciated, inequality.
منابع مشابه
Some rank equalities for finitely many tripotent matrices
A rank equality is established for the sum of finitely many tripotent matrices via elementary block matrix operations. Moreover, by using this equality and Theorems 8 and 10 in [Chen M. and et al. On the open problem related to rank equalities for the sum of finitely many idempotent matrices and its applications, The Scientific World Journal 2014 (2014), Article ID 702413, 7 page...
متن کاملThe Range of a Simple Random Walk on Z: An Elementary Combinatorial Approach
Two different elementary approaches for deriving an explicit formula for the distribution of the range of a simple random walk on Z of length n are presented. Both of them rely on Hermann Weyl’s discrepancy norm, which equals the maximal partial sum of the elements of a sequence. By this the original combinatorial problem on Z can be turned into a known path-enumeration problem on a bounded lat...
متن کاملA Recursive Approximation Approach of non-iid Lognormal Random Variables Summation in Cellular Systems
Co-channel interference is a major factor in limiting the capacity and link quality in cellular communications. As the co-channel interference is modeled by lognormal distribution, sum of the co-channel interferences of neighboring cells is represented by the sum of lognormal Random Variables (RVs) which has no closed-form expression. Assuming independent, identically distributed (iid) RVs, the...
متن کاملCartesian decomposition of matrices and some norm inequalities
Let X be an n-square complex matrix with the Cartesian decomposition X = A + i B, where A and B are n times n Hermitian matrices. It is known that $Vert X Vert_p^2 leq 2(Vert A Vert_p^2 + Vert B Vert_p^2)$, where $p geq 2$ and $Vert . Vert_p$ is the Schatten p-norm. In this paper, this inequality and some of its improvements ...
متن کاملSome inequalities involving lower bounds of operators on weighted sequence spaces by a matrix norm
Let A = (an;k)n;k1 and B = (bn;k)n;k1 be two non-negative ma-trices. Denote by Lv;p;q;B(A), the supremum of those L, satisfying the followinginequality:k Ax kv;B(q) L k x kv;B(p);where x 0 and x 2 lp(v;B) and also v = (vn)1n=1 is an increasing, non-negativesequence of real numbers. In this paper, we obtain a Hardy-type formula forLv;p;q;B(H), where H is the Hausdor matrix and 0 < q p 1. Also...
متن کامل