ON RANDOM WEIGHTED SUM OF POSITIVE SEMI-DEFINITE MATRICES

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Submitted: 2025-01-29; Published: 2025-01-29

This work is licensed under Creative Commons Attribution–NonCommercial International License (CC BY-NC 4.0).

Abstract

LetA1,...,Anbe fixed positive semi-definite matrices, i.e.Ai∈S+p(R)∀i∈{1,...,n}andu1,...,unare i.i.d. withui∼N(1,1). Then, the object ofour interest is the following probabilityP(n∑i=1uiAi∈S+p(R)).In this paper we examine this quantity for pairwise commutative matrices.Under some generic assumption about the matrices we prove that the weightedsum is also positive semi-definite with an overwhelming probability. Thisprobability tends to1exponentially fast by the growth of number of matricesnand is a linear function with respect to the matrix dimension p.