Euclidean Jordan algebras are the abstract foundation for symmetriccone optimization. Every element in a Euclidean Jordan algebra has a complete spectral decomposition analogous to the spectral decomposition of a real symmetric matrix into rank-one projections. The spectral decomposition in a Euclidean Jordan algebra stems from the likewise-analogous characteristic polynomial of its elements, whose 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.
Journal of Symbolic Computation, 2022