Mathematical Programming

, Volume 117, Issue 1–2, pp 195–221 | Cite as

Z-transformations on proper and symmetric cones

  • M. Seetharama GowdaEmail author
  • Jiyuan Tao


Motivated by the similarities between the properties of Z-matrices on \(R^{n}_+\) and Lyapunov and Stein transformations on the semidefinite cone \(\mathcal {S}^n_+\) , we introduce and study Z-transformations on proper cones. We show that many properties of Z-matrices extend to Z-transformations. We describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations. Finally, we study the equivalence of Q and P properties of Z-transformations on symmetric cones. In particular, we prove such an equivalence on the Lorentz cone.


Z-transformation Symmetric cone Quadratic representation Diagonal stability 

Mathematics Subject Classification (2000)

90C33 17C55 15A48 37B25 


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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of MarylandBaltimoreUSA
  2. 2.Department of Mathematical SciencesLoyola College in MarylandBaltimoreUSA

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