## Invited Session Tue.1.MA 041

#### Tuesday, 10:30 - 12:00 h, Room: MA 041

**Cluster 3: Complementarity & variational inequalities** [...]

### Complementarity properties of linear transformations on Euclidean Jordan algebras

**Chair: Jiyuan Tao**

**Tuesday, 10:30 - 10:55 h, Room: MA 041, Talk 1**

**Jeyaraman Irulappasamy**

P and semimonotonicity properties of linear transformations on Euclidean Jordan algebras

**Abstract:**

Let (*V, º, ⟨ ·, · ⟩ *) be a Euclidean Jordan algebra with the symmetric cone *K = {x º x : x ∈ V }.* Given a linear transformation *L : V → V* and *q ∈ V*,

the *linear complementarity problem* over the symmetric cone, LCP(*L, q*), is to find a vector *x ∈ V* such that *x ∈ K ,~ y:= L(x) + q ∈ K, * and * ⟨ x, y ⟩ = 0.* This problem includes the standard, semidefinite and second order linear complementarity problems. To study the existence and uniqueness of solution of the standard linear complementarity problem, several matrix classes have been introduced which includes *P* and semimonotone matrices. Motivated by these concepts, the matrix classes were extended to linear transformations on *S*^{n}, the space of all *n × n* real symmetric matrices, and further extended to Euclidean Jordan algebras. In this talk, we introduce various *P* and semimonotonicity properties and describe some interconnections between them. We also discuss how these concepts are significant in the study of LCP(*L, q*).

**Tuesday, 11:00 - 11:25 h, Room: MA 041, Talk 2**

**Jiyuan Tao**

The completely-Q property for linear transformations on Euclidean Jordan algebras

**Abstract:**

In this talk, we present a characterization of the completely-Q property for linear transformations on Euclidean Jordan algebras and show the completely-Q property and related properties on Euclidean Jordan algebras.

**Tuesday, 11:30 - 11:55 h, Room: MA 041, Talk 3**

**Roman Sznajder**

Complementarity properties of linear transformations on product spaces via Schur complements

**Coauthors: M. Seetharama Gowda, Jiyuan Tao**

**Abstract:**

In this paper we extend, in a natural way, the notion of the Schur complement of a subtransformation of a linear transformation defined on the product of two simple Euclidean Jordan algebras or, more generally, on two finite dimensional real Hilbert spaces. We study various complementarity properties of linear transformations in relations to subtransformations, principal pivot transformations, and Schur complements. We also investigate some relationships with dynamical systems.