Euclidean Jordan Algebra

A Euclidean Jordan algebra is a finite-dimensional real vector space with a commutative bilinear product satisfying the Jordan identity , together with an inner product for which each left multiplication is self-adjoint (). The prototype is the Hermitian matrices with and . The product is non-associative.

The quadratic representation is , ; on matrices . It is the Jordan-algebraic analogue of congruence and is defined for every , invertible or not.

Proposition. Peirce projection

For an idempotent (), has eigenvalues and splits into the Peirce decomposition . Then is the orthogonal projection onto (eigenvalues ), with kernel ; is a subalgebra and is a positive map. In matrix terms , is the off-diagonal block and the complementary diagonal block.

Theorem. Symmetric cone (Koecher–Vinberg)

The squares form a symmetric cone, self-dual and homogeneous. The Koecher–Vinberg theorem identifies the Euclidean Jordan algebras with exactly the coordinates of the symmetric cones, so the algebra and its cone carry the same information.

Remark. Positive semidefinite (PSD) versus SPD is associative versus Jordan

The PSD cone (eigenvalues ) is closed, boundary included, whereas the SPD interior (eigenvalues ) is open, a manifold and, under the affine-invariant metric, the symmetric space . The distinction is algebraic, not merely topological. The associative route treats as a homogeneous space of acting by congruence , which needs invertible and so cannot reach . The Jordan route uses , defined for singular , so it covers . Non-associativity is what buys the boundary.