Rank computation in Euclidean Jordan algebras. (November 2022)

• Record Type: Journal Article
• Authors: Orlitzky, Michael
• Is Part Of: Journal of symbolic computation. Volume 113(2022)
• Page Start: 181
• Page End: 192
• Publication Date: 2022-11
• Subjects: 17–08 -- 17C99 -- 13P15 -- 15A06 -- 15A54; Euclidean Jordan algebra -- Characteristic polynomial -- Spectral decomposition -- Rank -- Symmetric cone; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2022.04.003
• Languages: English
• ISSNs: 0747-7171

Abstract

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.