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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2604.04650 |
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| _version_ | 1866915917835796480 |
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| author | Vrabel, Robert |
| author_facet | Vrabel, Robert |
| contents | This paper develops a unified analytical framework for determinant identities under finite-rank perturbations of square matrices that remains valid without invertibility assumptions. In contrast to classical inverse-based formulations, the approach is based on an adjugate-driven additive representation, which extends naturally to singular matrices and yields explicit, non-asymptotic formulas. Building on this representation, we derive recursive and multiplicative expressions describing the evolution of determinant and log-determinant quantities under successive rank-one updates. These results reveal a structural interpretation in which determinant-based quantities evolve as cumulative measures of independent directions, providing a precise decomposition of incremental contributions. To address the singular case, we develop a systematic extension based on the Drazin inverse and the pseudodeterminant, leading to closed-form identities that isolate the contribution of the nonzero spectrum. In particular, we obtain a generalized determinant formula that can be viewed as a singular counterpart of the matrix determinant lemma. The spectral impact of low-rank perturbations is analyzed, yielding explicit conditions governing eigenvalue shifts and stability preservation. The proposed framework establishes a direct analytical link between matrix perturbation theory and system-theoretic concepts. In particular, we show that the pseudodeterminant of controllability Gramians admits a multiplicative decomposition that explicitly quantifies the incremental expansion of the reachable subspace under successive inputs. This leads to a unified interpretation of information accumulation, uncertainty reduction, and reachability in both full-rank and rank-deficient linear systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_04650 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Determinant Dynamics under Low-Rank Perturbations: A Unified Framework for Singular Systems Vrabel, Robert Optimization and Control 15A15 15A09 93B05 This paper develops a unified analytical framework for determinant identities under finite-rank perturbations of square matrices that remains valid without invertibility assumptions. In contrast to classical inverse-based formulations, the approach is based on an adjugate-driven additive representation, which extends naturally to singular matrices and yields explicit, non-asymptotic formulas. Building on this representation, we derive recursive and multiplicative expressions describing the evolution of determinant and log-determinant quantities under successive rank-one updates. These results reveal a structural interpretation in which determinant-based quantities evolve as cumulative measures of independent directions, providing a precise decomposition of incremental contributions. To address the singular case, we develop a systematic extension based on the Drazin inverse and the pseudodeterminant, leading to closed-form identities that isolate the contribution of the nonzero spectrum. In particular, we obtain a generalized determinant formula that can be viewed as a singular counterpart of the matrix determinant lemma. The spectral impact of low-rank perturbations is analyzed, yielding explicit conditions governing eigenvalue shifts and stability preservation. The proposed framework establishes a direct analytical link between matrix perturbation theory and system-theoretic concepts. In particular, we show that the pseudodeterminant of controllability Gramians admits a multiplicative decomposition that explicitly quantifies the incremental expansion of the reachable subspace under successive inputs. This leads to a unified interpretation of information accumulation, uncertainty reduction, and reachability in both full-rank and rank-deficient linear systems. |
| title | Determinant Dynamics under Low-Rank Perturbations: A Unified Framework for Singular Systems |
| topic | Optimization and Control 15A15 15A09 93B05 |
| url | https://arxiv.org/abs/2604.04650 |