2020; Physical and Mathematical Sciences, 54(2 (252): 96–100
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ON RANDOM WEIGHTED SUM OF POSITIVE SEMI-DEFINITE MATRICES

Tigran Galstyan*

Received: 2025-01-29 · Published: 2020-08-17

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CC BY-NC 4.0 This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
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Original title
ON RANDOM WEIGHTED SUM OF POSITIVE SEMI-DEFINITE MATRICES
Author
Tigran Galstyan
Published
2020-08-17
Licence
Creative Commons Attribution-NonCommercial 4.0 International
Original
https://doi.org/10.46991/PYSU:A/2020.54.2.096

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.
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