, Volume 82, Issue 1, pp 159-167

Yuan's alternative theorem and the maximization of the minimum eigenvalue function

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Abstract

LetA 1 andA 2 be two symmetric matrices of ordern×n. According to Yuan, there exists a convex combination of these matrices which is positive semidefinite, if and only if the functionxR n ↦ max {x T A 1 x,x T A 2 x} is nonnegative. We study the case in which more than two matrices are involved. We study also a related question concerning the maximization of the minimum eigenvalue of a convex combination of symmetric matrices.

Communicated by F. Giannessi
This research was partially supported by Dirección General de Investigación Científica y Técnica (DGICYT) under Project PB92-0615.