Abstract
It is well known thatσ(H), the sum of the negative eigenvalues of a Hermitian matrixH, is a concave and increasing function ofH. In contrast to this, we prove that forA nonsingular Hermitian andP positive definite, the functionP↦σ(AP)=σ(P 1/2 AP 1/2) is convex and decreasing. Several other results of this nature are also proved.
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Lieb, E.H., Siedentop, H. Convexity and concavity of eigenvalue sums. J Stat Phys 63, 811–816 (1991). https://doi.org/10.1007/BF01029984
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DOI: https://doi.org/10.1007/BF01029984