Weak interlacing properties of totally positive matrices

@article{Friedland1985WeakIP,
  title={Weak interlacing properties of totally positive matrices},
  author={Shmuel Friedland},
  journal={Linear Algebra and its Applications},
  year={1985},
  volume={71},
  pages={95-100}
}
  • S. Friedland
  • Published 1 November 1985
  • Mathematics
  • Linear Algebra and its Applications
Interlacing inequalities for totally nonnegative matrices
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TLDR
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In a previous paper we proved that the diagonal elements of a totally nonnegative matrix are majorized by its eigenvalues. In this note we show that the majorization of a vector of nonnegative real
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LetA1,...,An andK bem×m symmetric matrices withK positive definite. Denote byC the convex hull of {A1,...An}. Let {λp(KA)}1n be then real eigenvalues ofKA arranged in decreasing order. We show that
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1) Spec A[Jl.l n IR =1= t/>, for t/> c Jl. S (n), 2) I(A[J-L]) « I(A[v]), if t/> c v S Jl. S (n), where I(A[Jl.]) = min(Spec A[Jl.l n IR). For A, BE W(n), define A «, B by I(A[J-L]) « I(B[J-L]), for