A Unique Inverse Decomposition of Positive Definite Matrices under Linear Constraints
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866911399538589696 |
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| author | Dolinsky, Yan Zuk, Or |
| author_facet | Dolinsky, Yan Zuk, Or |
| contents | 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 inverse component is required to belong to 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 \emph{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 show that the proposed decomposition arises naturally in exponential utility maximization, a central problem in mathematical finance. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_18662 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Unique Inverse Decomposition of Positive Definite Matrices under Linear Constraints Dolinsky, Yan Zuk, Or Optimization and Control Numerical Analysis 15A39, 15B48, 90C25 (primary), 91G10 (secondary) 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 inverse component is required to belong to 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 \emph{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 show that the proposed decomposition arises naturally in exponential utility maximization, a central problem in mathematical finance. |
| title | A Unique Inverse Decomposition of Positive Definite Matrices under Linear Constraints |
| topic | Optimization and Control Numerical Analysis 15A39, 15B48, 90C25 (primary), 91G10 (secondary) |
| url | https://arxiv.org/abs/2601.18662 |