pith. machine review for the scientific record. sign in

arxiv: 2604.04650 · v1 · submitted 2026-04-06 · 🧮 math.OC

Recognition: no theorem link

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

Robert Vrabel
Authors on Pith no claims yet

Pith reviewed 2026-05-10 19:47 UTC · model grok-4.3

classification 🧮 math.OC
keywords determinant identitieslow-rank perturbationssingular matricesadjugatepseudodeterminantDrazin inversecontrollability Gramianmatrix determinant lemma
0
0 comments X

The pith

An adjugate-driven representation supplies explicit determinant formulas for low-rank perturbations of singular matrices.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a framework for determinant identities under finite-rank perturbations that avoids any invertibility requirement on the matrix. It replaces classical inverse-based formulas with an adjugate-driven additive representation, which produces recursive and multiplicative update rules for both the determinant and the log-determinant. These rules interpret the determinant as a running tally of independent directions contributed by each perturbation. In the singular setting the same representation, combined with the Drazin inverse and the pseudodeterminant, isolates the nonzero spectrum and yields a direct generalization of the matrix determinant lemma. The framework further shows that the pseudodeterminant of a controllability Gramian factors multiplicatively, thereby measuring the incremental growth of the reachable subspace under successive inputs.

Core claim

The central claim is that the determinant under finite-rank perturbations admits an adjugate-driven additive representation valid for any square matrix, yielding recursive expressions under successive rank-one updates and a generalized determinant identity via the Drazin inverse and pseudodeterminant that serves as the singular counterpart of the matrix determinant lemma; the same machinery produces a multiplicative decomposition of the pseudodeterminant of controllability Gramians that quantifies reachable-subspace expansion.

What carries the argument

The adjugate-driven additive representation, which expresses the change in determinant as an additive correction involving the adjugate matrix rather than an inverse.

If this is right

  • Recursive and multiplicative formulas describe the evolution of determinant and log-determinant under successive rank-one updates.
  • Determinant-based quantities serve as cumulative measures of the independent directions contributed by each perturbation.
  • Explicit conditions govern eigenvalue shifts and stability preservation under low-rank perturbations.
  • The pseudodeterminant of controllability Gramians admits a multiplicative decomposition that quantifies incremental expansion of the reachable subspace.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This decomposition could be used to track information accumulation in rank-deficient linear systems without requiring full-rank assumptions.
  • The framework suggests a direct route for analyzing stability margins in singular control problems by monitoring pseudodeterminant growth.
  • Verification on low-dimensional discrete-time examples would test whether the multiplicative decomposition accurately predicts reachable-subspace volume after each input.

Load-bearing premise

The adjugate-driven additive representation holds for arbitrary square matrices without further restrictions on structure or perturbation rank.

What would settle it

Direct numerical evaluation of the proposed generalized determinant formula on a small singular matrix after a known rank-one update, compared against the exact pseudodeterminant of the perturbed matrix.

Figures

Figures reproduced from arXiv: 2604.04650 by Robert Vrabel.

Figure 1
Figure 1. Figure 1: Starting from the regularized matrix W0 = εI, the associated ellipsoid {x ∈ R n : x TW−1 0 x ≤ 1} represents the set of states reachable with bounded input energy. The normalization by 1 fixes the energy level and ensures that the volume of the set is directly related to the determinant. In this case, the ellipsoid reduces to the Euclidean ball {x ∈ R n : ∥x∥ ≤ √ ε}, that is, a circle of radius √ ε in two … view at source ↗
Figure 1
Figure 1. Figure 1: Evolution of the controllability Gramian ellipsoid for the discrete-time sys [PITH_FULL_IMAGE:figures/full_fig_p029_1.png] view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript develops a unified analytical framework for determinant identities under finite-rank perturbations of square matrices that holds without invertibility assumptions. It relies on an adjugate-driven additive representation to derive recursive and multiplicative expressions for the evolution of determinants and log-determinants under successive rank-one updates. For singular matrices, the framework extends these results using the Drazin inverse and pseudodeterminant to obtain closed-form identities that isolate the nonzero spectrum contribution, including a generalized determinant formula presented as a singular counterpart to the matrix determinant lemma. Spectral analysis of eigenvalue shifts under low-rank perturbations is provided, along with an application to control theory showing that the pseudodeterminant of controllability Gramians admits a multiplicative decomposition quantifying incremental expansion of the reachable subspace under successive inputs.

Significance. If the derivations are correct and the assumptions hold, the work offers explicit non-asymptotic formulas linking matrix perturbation theory to system reachability in both full-rank and rank-deficient linear systems. The parameter-free character of the adjugate-based approach and the direct quantification of reachable-subspace growth are potentially useful for controllability analysis and uncertainty reduction in singular systems. The extension to pseudodeterminants via the Drazin inverse addresses a gap in classical inverse-based identities.

major comments (2)
  1. [Abstract / singular-case section] Abstract and the section developing the singular extension: the claim that the adjugate-driven additive representation extends naturally to singular matrices without additional restrictions on matrix structure or perturbation rank is load-bearing for the generalized determinant formula and the multiplicative decomposition of pseudodeterminants. However, because A adj(A) = 0 for singular A, any additive update must handle cases where the low-rank perturbation has a nontrivial component in the kernel; this can alter the index or algebraic multiplicity and prevent clean isolation of the nonzero spectrum by the Drazin inverse. The manuscript should provide an explicit counter-example check or additional hypothesis (e.g., range condition on the perturbation) to confirm the formulas remain valid.
  2. [Application to controllability Gramians] The application to controllability Gramians: the multiplicative decomposition of the pseudodeterminant is presented as quantifying incremental expansion of the reachable subspace. This step relies on the preceding singular identities holding without kernel-interaction caveats; if the first major comment is not resolved, the system-theoretic interpretation loses its claimed explicitness.
minor comments (2)
  1. Notation for the pseudodeterminant and Drazin inverse should be introduced with a brief reminder of their definitions and domains of applicability early in the manuscript to aid readers unfamiliar with singular-matrix theory.
  2. The recursive formulas for successive rank-one updates would benefit from a small numerical example (e.g., 3-by-3 singular matrix with explicit rank-one updates) to illustrate the multiplicative decomposition before the general proof.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for raising these important points regarding the singular extension. We address each major comment below and indicate the revisions we intend to make.

read point-by-point responses

  1. Referee: [Abstract / singular-case section] Abstract and the section developing the singular extension: the claim that the adjugate-driven additive representation extends naturally to singular matrices without additional restrictions on matrix structure or perturbation rank is load-bearing for the generalized determinant formula and the multiplicative decomposition of pseudodeterminants. However, because A adj(A) = 0 for singular A, any additive update must handle cases where the low-rank perturbation has a nontrivial component in the kernel; this can alter the index or algebraic multiplicity and prevent clean isolation of the nonzero spectrum by the Drazin inverse. The manuscript should provide an explicit counter-example check or additional hypothesis (e.g., range condition on the perturbation) to confirm the formulas remain valid.

    Authors: We appreciate the referee highlighting this subtlety in the singular case. The adjugate identity A adj(A) = det(A) I holds for every square matrix, and our additive representation is derived directly from this relation without invoking invertibility. The generalized determinant formula and pseudodeterminant identities are obtained by projecting onto the nonzero spectrum via the Drazin inverse, which is well-defined for any square matrix. While a perturbation component in the kernel can change the index or Jordan structure, the pseudodeterminant itself depends only on the nonzero eigenvalues, and the update formulas track their evolution algebraically through the adjugate terms. To strengthen the presentation, we will add a short remark in the singular-case section together with an explicit low-dimensional counter-example verification (a 2-by-2 singular matrix with a kernel-interacting rank-one update) that confirms the formulas continue to isolate the nonzero spectrum correctly. If the example reveals a boundary case, we will state the mild range condition under which the clean isolation holds. revision: partial

  2. Referee: [Application to controllability Gramians] The application to controllability Gramians: the multiplicative decomposition of the pseudodeterminant is presented as quantifying incremental expansion of the reachable subspace. This step relies on the preceding singular identities holding without kernel-interaction caveats; if the first major comment is not resolved, the system-theoretic interpretation loses its claimed explicitness.

    Authors: The controllability-Gramian application follows directly from the singular-case identities. With the additional verification example and clarifying remark added in response to the first comment, the multiplicative decomposition of the pseudodeterminant remains valid and continues to quantify the incremental expansion of the reachable subspace under successive inputs. We will revise the application section to include an explicit cross-reference to the new singular-case verification, thereby ensuring the system-theoretic claims rest on fully supported identities. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivations from adjugate and Drazin inverse are self-contained

full rationale

The paper builds its framework on the adjugate-driven additive representation and extensions via the Drazin inverse and pseudodeterminant to handle singular matrices, presenting these as direct extensions of the classical matrix determinant lemma without any reduction to fitted parameters, self-definitional loops, or load-bearing self-citations. No equations or claims in the abstract or description reduce the central results (generalized determinant formulas, multiplicative decompositions for pseudodeterminants of Gramians) to their own inputs by construction. The approach is positioned as analytically independent, relying on standard linear algebra identities that remain externally verifiable.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard properties of the adjugate matrix, Drazin inverse, and pseudodeterminant from linear algebra. No new free parameters or invented entities are apparent from the abstract. The main contribution appears to be unification and extension rather than new postulates.

axioms (2)
  • standard math The adjugate matrix satisfies identities allowing an additive representation of determinant changes under perturbations.
    Invoked as the basis for the framework that remains valid without invertibility assumptions.
  • standard math The Drazin inverse and pseudodeterminant are well-defined for square matrices and isolate the contribution of the nonzero spectrum.
    Used to develop the extension to singular matrices and the generalized determinant formula.

pith-pipeline@v0.9.0 · 5546 in / 1578 out tokens · 85823 ms · 2026-05-10T19:47:57.057087+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

16 extracted references · 11 canonical work pages

  1. [1]

    R. A. Horn, C. R. Johnson, Matrix Analysis, Cambridge University Press, 1985. 30

    1985

  2. [2]

    D. A. Harville, Matrix Algebra From a Statistician’s Perspective, Springer, 1997

    1997

  3. [3]

    G. J. Bierman, Sequential square root filtering and smoothing of discrete linear systems, Automatica 10 (2) (1974) 147–158. doi:https://doi.org/10.1016/0005- 1098(74)90020-X

    work page doi:10.1016/0005- 1974

  4. [4]

    Verhaegen, P

    M. Verhaegen, P. Van Dooren, Numerical aspects of different Kalman filter implementations, IEEE Transactions on Automatic Control 31 (10) (1986) 907–

    1986

  5. [5]

    doi:10.1109/TAC.1986.1104128

    work page doi:10.1109/tac.1986.1104128 1986

  6. [6]

    Schmidt, P

    J. Schmidt, P. Hennig, J. Nick, F. Tronarp, The rank-reduced Kalman filter: Approximate dynamical-low-rank filtering in high dimensions, in: Advances in Neural Information Processing Systems (NeurIPS), Vol. 36, New Orleans, Louisiana, USA, 2023, pp. 1–13

    2023

  7. [7]

    Y. Li, B. Wahlberg, X. Hu, L. Xie, Inverse Kalman filtering problems for discrete-time systems, Automatica 163 (2024) 111560. doi:https://doi.org/10.1016/j.automatica.2024.111560

    work page doi:10.1016/j.automatica.2024.111560 2024

  8. [8]

    S. Boyd, L. Vandenberghe, Convex Optimization, Cambridge, 2004. doi:https://doi.org/10.1017/CBO9780511804441

    work page doi:10.1017/cbo9780511804441 2004

  9. [9]

    Z. Kang, C. Peng, J. Cheng, Q. Cheng, Logdet rank minimization with application to subspace clustering, Computational Intelligence and Neuroscience 2015 (1) (2015) 824289. doi:https://doi.org/10.1155/2015/824289

    work page doi:10.1155/2015/824289 2015

  10. [10]

    Kailath, Linear Systems, Prentice-Hall, Inc., 1980

    T. Kailath, Linear Systems, Prentice-Hall, Inc., 1980

    1980

  11. [11]

    J. E. Gough, G. Zhang, On realization theory of quantum linear systems, Automatica 59 (2015) 139–151. doi:https://doi.org/10.1016/j.automatica.2015.06.023

    work page doi:10.1016/j.automatica.2015.06.023 2015

  12. [12]

    Olfati-Saber, Distributed Kalman filtering for sensor networks, in: 2007 46th IEEE Conference on Decision and Control, 2007, pp

    R. Olfati-Saber, Distributed Kalman filtering for sensor networks, in: 2007 46th IEEE Conference on Decision and Control, 2007, pp. 5492–5498. doi:10.1109/CDC.2007.4434303

    work page doi:10.1109/cdc.2007.4434303 2007

  13. [13]

    M. P. Drazin, Pseudo-inverses in associative rings and semigroups, The American Mathematical Monthly 65 (7) (1958) 506–514. doi:10.1080/00029890.1958.11991949

    work page doi:10.1080/00029890.1958.11991949 1958

  14. [14]

    Vrabel, A note on the matrix determinant lemma, International Journal of Pure and Applied Mathematics 111 (4) (2016) 643–646

    R. Vrabel, A note on the matrix determinant lemma, International Journal of Pure and Applied Mathematics 111 (4) (2016) 643–646. doi:10.12732/ijpam.v111i4.11

    work page doi:10.12732/ijpam.v111i4.11 2016

  15. [15]

    T. M. Cover, J. A. Thomas, Elements of Information Theory, 2nd Edition, John Wiley & Sons, Inc., 2006. doi:10.1002/047174882X

    work page doi:10.1002/047174882x 2006

  16. [16]

    A. G. Akritas, E. K. Akritas, G. I. Malaschonok, Various proofs of Sylvester’s (determinant) identity, Mathematics and Computers in Simulation 42 (4) (1996) 585–593. doi:https://doi.org/10.1016/S0378-4754(96)00035-3. 31

    work page doi:10.1016/s0378-4754(96)00035-3 1996