Optimization Letters

, Volume 8, Issue 3, pp 939–947 | Cite as

Positive definite matrix approximation with condition number constraint

  • Mirai TanakaEmail author
  • Kazuhide Nakata
Original Paper


Positive definite matrix approximation with a condition number constraint is an optimization problem to find the nearest positive definite matrix whose condition number is smaller than a given constant. We demonstrate that this problem can be converted to a simpler one when we use a unitary similarity invariant norm as a metric. We can especially convert it to a univariate piecewise convex optimization problem when we use the Ky Fan p-k norm. We also present an analytical solution to the problem whose metric is the spectral norm and the trace norm.


Matrix approximation Covariance estimation Unitary similarity invariant norm Ky Fan p-k norm 



The second author was supported by Grant-in-Aid for Young Scientists (B) 22710136.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Ookayama, Meguro-kuJapan

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