Euclidean Jordan Algebra

Although the setting for conjugate and perturbation duality is Euclidean space, many important applications involve additional structure. Whereas applied optimization theory commonly adds this extra structure informally, we proceed formally, by working in a “Euclidean Jordan algebra,” a Euclidean space having additional algebraic structure. In mathematics, an algebra is a vector space with a bilinear, multiplicative product. A Euclidean Jordan algebra is a finite-dimensional real algebra satisfying certain conditions. The multiplicative product is the Jordan product, written x•y. Given a finite-dimensional real vector space, to define the Jordan product specifies the structure of the Euclidean Jordan algebra. The prime example of a Euclidean Jordan algebra is the self-adjoint linear transformations, in which the Jordan product $${\textit {{\sf { {X}}}}}\bullet {\textit {{\sf { {Y}}}}}:= \frac {1}{2}({\textit {{\sf { {X} {Y}}}}}+ {\textit {{\sf { {Y} {X}}}}}).$$ That the product is commutative but not associative is a key property, the reverse of standard matrix multiplication. We present the original axioms of Jordan, von Neumann, and Wigner, which capture very well how a Euclidean Jordan algebra generalizes the real numbers, but not the complex numbers. By definition, x⪰y means that x−y belongs to the cone of squares—all vectors equal to the square of another. Whereas there is only one type of Euclidean space, in that any two Euclidean spaces of the same dimension are isomorphic, in contrast the classification theorem states that there are five distinct types of Euclidean Jordan algebras.