Information Geometry of Asymmetric Interaction Matrices

Asymmetric interaction matrices encode the linear coupling structure and directed interaction patterns that arise in mathematical models of complex networks across ecology, finance, and machine learning, yet their geometric structure as points on a statistical manifold has received comparatively little systematic treatment. This paper develops a rigorous information-geometric framework for the manifold Mn+ of real n×n interaction matrices whose symmetric part is negative definite—equivalently, the matrices satisfying the numerical stability condition ω(A)=λmax(S(A))<0. The symmetric part S(A)=(A+AT)/2 and the skew-symmetric part K(A)=(A−AT)/2 correspond, respectively, to the metric structure and the torsion of the induced statistical manifold. We construct the natural augmented Riemannian metric g on Mn+ as the sum of the Fisher–Rao pullback metric through −S(·) and a Frobenius term on K(·), derive explicit formulae for the sectional curvature in the mixed symmetric–skew plane, and prove that the sectional curvature vanishes if and only if A is normal. The central theoretical result is a curvature-mediated stability theorem: a Fisher–Rao stability margin, derived from the precision representative of A, provides a sharp, computationally accessible certificate for the asymptotic stability of the linear dynamical system x˙=Ax, with the instability boundary lying at infinite Fisher–Rao distance. We further establish an information-geometric reformulation of May’s stability criterion for random ecological networks, a curvature-based covariance regularisation scheme for financial correlation matrices, and a Jacobian stability bound for deep neural networks. All the main results are illustrated with explicit 3×3 and 4×4 numerical examples.