Rank computation in Euclidean Jordan algebras

Abstract

Euclidean Jordan algebras are the abstract foundation for symmetric cone optimization. Every element in a Euclidean Jordan algebra has a complete spectral decomposition analogous to and subsuming that of a real symmetric matrix into rank-one projections. This general spectral decomposition stems from the element's likewise-analogous characteristic polynomial whose degree (they all have the same degree) is called the rank of the algebra. As a prerequisite for the spectral decomposition, we derive an algorithm that computes the rank of a Euclidean Jordan algebra and allows us to construct the characteristic polynomials of its elements. The ultimate goal of this work is to support a generic computational framework for solving symmetric cone optimization problems in Jordan-algebraic terms.