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Extremal eigenvalue problems for convex sets of symmetric matrices and operators

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Abstract

LetA 1,...,An andK bem×m symmetric matrices withK positive definite. Denote byC the convex hull of {A 1,...An}. Let {λ p (KA)} n1 be then real eigenvalues ofKA arranged in decreasing order. We show that maxλ p (KA) onC is attained for someA *=Σ = 1/n i for which at mostp(p+1)/2 of α i * do not vanish. We extend this result in several directions and consider applications to classes of integral equations.

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Authors and Affiliations

  1. Technion-Israel Institute of Technology, Haifa, Israel

    Shmuel Friedland

  2. The Weizmann Institute of Science, Rehovot, Israel

    Shmuel Friedland

Authors
  1. Shmuel Friedland
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Additional information

This paper is based mainly on the author’s doctoral dissertation written at the Technion—Israel Institute of Technology, March 1971, under the direction of Professor B. Schwarz. I wish to thank Professor Schwarz for his advice and encouragement. I am also grateful to Professor S. Karlin for supplying simplifications of several of my arguments. Some extensions discussed here are joint results of Karlin and the author.

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Friedland, S. Extremal eigenvalue problems for convex sets of symmetric matrices and operators. Israel J. Math. 15, 311–331 (1973). https://doi.org/10.1007/BF02787574

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  • DOI: https://doi.org/10.1007/BF02787574

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