Complementarity properties of Peirce-diagonalizable linear transformations on Euclidean Jordan algebras^†

Abstract

Peirce-diagonalizable linear transformations on a Euclidean Jordan algebra are of the form L(x)=A·x:=∑ a _ij x _ij , where A=[a _ij ] is a real symmetric matrix and ∑ x _ij is the Peirce decomposition of an element x in the algebra with respect to a Jordan frame. Examples of such transformations include Lyapunov transformations and quadratic representations on Euclidean Jordan algebras. Schur (or Hadamard) product of symmetric matrices provides another example. Motivated by a recent generalization of the Schur product theorem, we study general and complementarity properties of such transformations.

Keywords:

• Peirce-diagonalizable transformation
• linear complementarity problem
• Euclidean Jordan algebra
• symmetric cone
• Schur/Hadamard product
• Lypaunov transformation
• quadratic representation

2010 AMS Subject Classifications :

• 15A33
• 17C20
• 17C65
• 90C33

Acknowledgements

We thank the referees for their constructive comments which improved and simplified the presentation of the paper. In particular, we thank one referee for Lemma 2.4 and suggestions which lead to the explicit formulation of Proposition 2.3 and simplification of various proofs.

Notes

^†We dedicate this paper to our friend, colleague,, teacher Florian Potra on the occasion of his 60th birthday.