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Determinant Dynamics under Low-Rank Perturbations: A Unified Framework for Singular Systems

Robert Vrabel

Abstract

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.

Determinant Dynamics under Low-Rank Perturbations: A Unified Framework for Singular Systems

Abstract

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.

Paper Structure

This paper contains 13 sections, 97 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notations and preliminaries
  3. Finite-rank perturbations and determinant dynamics
  4. Log-determinant representation
  5. A singular extension with the Drazin inverse
  6. Spectral implications of low-rank perturbations
  7. Applications to estimation and control
  8. Covariance updates and information accumulation
  9. Connection to Kalman-type updates
  10. A singular $3 \times 3$ example: control and information perspective
  11. Gramian determinants and reachability
  12. Directional novelty and robustness under perturbations
  13. Conclusions

Figures (1)

  • Figure 1: Evolution of the controllability Gramian ellipsoid for the discrete-time system $x_{t+1}=Ax_t+Bu_t$ with $A=\left[0.720.55-0.180.78\blacktriangleleft\blacktriangleleft\newline\right]$, $B=\left[1.00.15\blacktriangleleft\blacktriangleleft\newline\right]$, and horizon $N=4$. Starting from the regularized matrix $W_0=\varepsilon I$, the ellipsoid $\{x : x^T W_k^{-1} x \le 1\}$ evolves under successive rank-one contributions $u_\ell u_\ell^T$, where, for $m=1$, the update directions are given by $u_\ell = A^{\ell-1} B$, $\ell=1,\dots,N$. Each update expands the ellipsoid in directions not yet represented in the Gramian, while contributions aligned with existing directions induce only marginal growth. This provides a geometric interpretation of the multiplicative structure of $\mathop{\rm pdet}\nolimits(W_k)$ and the incremental expansion of the reachable set.