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Z-transformations on proper and symmetric cones

Z-transformations

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Abstract

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.

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Correspondence to M. Seetharama Gowda.

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This paper is dedicated to Professor Steve Robinson on the occasion of his 65th birthday. His ideas and results greatly influenced a generation of researchers including the first author of this paper.

Steve, thanks for being a great teacher, mentor, and a friend. Best wishes for a long and healthy productive life.

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Gowda, M.S., Tao, J. Z-transformations on proper and symmetric cones. Math. Program. 117, 195–221 (2009). https://doi.org/10.1007/s10107-007-0159-8

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  • DOI: https://doi.org/10.1007/s10107-007-0159-8

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