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Numerische Mathematik

, Volume 69, Issue 1, pp 17–23 | Cite as

Minimizing the condition number of a positive definite matrix by completion

  • L. Elsner
  • C. He
  • V. Mehrmann

Summary.

We consider the problem of minimizing the spectral condition number of a positive definite matrix by completion:\(\min\left\{ {\rm cond}\left(\mat{cc} A & B^{\rm H} \\ B & X \rix\right): \mat{cc} A & B^{\rm H} \\ B & X \rix \mbox {\rm positive definite} \right\},\) \noindent where \(A\) is an \(n\times n\) Hermitian positive definite matrix, \(B\) a \(p\times n\) matrix and \(X\) is a free \(p\times p\) Hermitian matrix. We reduce this problem to an optimization problem for a convex function in one variable. Using the minimal solution of this problem we characterize the complete set of matrices that give the minimum condition number.

Mathematics Subject Classification (1991):65F35, 15A12 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1994

Authors and Affiliations

  • L. Elsner
    • 1
  • C. He
    • 2
  • V. Mehrmann
    • 2
  1. 1.Fakult{\"a}t f{\"u}r Mathematik, Universit{\"a}t Bielefeld, D-33615 Bielefeld, Germany DE
  2. 2.Fachbereich Mathematik, PSF 964, TU-Chemnitz-Zwickau, D-09009 Chemnitz, Germany DE

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