SCIENCE@home

The norm of polynomials in large random and deterministic matrices

C. Male

Published 2010 in Probability Theory and Related Fields

ABSTRACT

Let $${{\bf X}_N =(X_1^{(N)}, \ldots, X_p^{(N)})}$$ be a family of N × N independent, normalized random matrices from the Gaussian Unitary Ensemble. We state sufficient conditions on matrices $${{\bf Y}_N =(Y_1^{(N)}, \ldots, Y_q^{(N)})}$$ , possibly random but independent of XN, for which the operator norm of $${P({\bf X}_N, {\bf Y}_N, {\bf Y}_N^*)}$$ converges almost surely for all polynomials P. Limits are described by operator norms of objects from free probability theory. Taking advantage of the choice of the matrices YN and of the polynomials P, we get for a large class of matrices the “no eigenvalues outside a neighborhood of the limiting spectrum” phenomena. We give examples of diagonal matrices YN for which the convergence holds. Convergence of the operator norm is shown to hold for block matrices, even with rectangular Gaussian blocks, a situation including non-white Wishart matrices and some matrices encountered in MIMO systems.

PUBLICATION RECORD

CITATION MAP

EXTRACTION MAP

CLAIMS

  • No claims are published for this paper.

CONCEPTS

  • No concepts are published for this paper.

REFERENCES

Showing 1-43 of 43 references · Page 1 of 1

CITED BY

Showing 1-75 of 75 citing papers · Page 1 of 1