The Cone of J-Hermitian Matrices and a Geometric Mean

We study the cone ${\mathcal{P}}_{\text{J}}$ of positive J-Hermitian matrices associated with an indefinite signature matrix J = ${\text{Id}}_{p,q}$. We show that the J-exponential map is bijective and use it to analyze the algebraic and geometric structure of ${\mathcal{P}}_{\text{J}}$. Through a canonical identification with the cone of positive definite matrices, we endow ${\mathcal{P}}_{\text{J}}$ with a natural Riemannian structure. In this setting, we define a J-geometric mean as the midpoint of geodesics and prove that it is uniquely characterized as the solution of a Riccati-type equation.