A Unique Inverse Decomposition of Positive Definite Matrices under Linear Constraints

Abstract

We study a nonlinear decomposition of a positive definite matrix into two components: the inverse of another positive definite matrix and a symmetric matrix constrained to lie in a prescribed linear subspace. Equivalently, the matrix being inverted is required to lie in the orthogonal complement of that subspace with respect to the trace inner product. Under a sharp nondegeneracy condition on the subspace, we show that every positive definite matrix admits a unique decomposition of this form. This decomposition admits a variational characterization as the unique minimizer of a strictly convex log-determinant optimization problem, which in turn yields a natural dual formulation that can be efficiently exploited computationally. We derive several properties, including the stability of the decomposition. We further develop feasibility-preserving Newton-type algorithms with provable convergence guarantees and analyze their per-iteration complexity in terms of algebraic properties of the decomposed matrix and the underlying subspace. Finally, we present two applications: one in mathematical finance, where the decomposition arises naturally in exponential utility maximization, and one in linear algebra, where our main theorem characterizes a new bilinear triangular decomposition for positive definite matrices, obtained together with a direct elimination algorithm.