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

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.