Pith. sign in

REVIEW 8 minor 30 references

Dagger-Drazin Inverses

T0 review · 0 major / 8 minor · reviewed 2026-08-08 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read The paper introduces dagger-Drazin inverses for arbitrary maps and proves they exist exactly when the induced positive maps are Drazin, making Moore-Penrose inverses the index-one case.

desk verdict Introduces a genuinely new generalized inverse for arbitrary maps in dagger categories, proves the advertised equivalences cleanly, and deserves a serious referee. read the letter →

arxiv 2502.05306 v3 pith:CSEODKRR submitted 2025-02-07 math.CT quant-ph

Robin Cockett (University of Calgary) , Jean-Simon Pacaud Lemay (Macquarie University) , Priyaa Varshinee Srinivasan (Tallinn University of Technology) This is my paper · ORCID
classification math.CTquant-ph MSC 18B9915A09
keywords daggercategoriesdagger-DrazininverseDrazinMoore-Penrosepositivemapsgeneralizedinversesopposingpairscofree
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Drazin inverses are a classical generalized inverse defined for endomorphisms—maps from an object to itself—and until now had no natural counterpart for arbitrary maps. This paper introduces the dagger-Drazin inverse for arbitrary maps in dagger categories, where the dagger $\dagger$ is an involutive adjoint operation. Its main theorem proves a direct reduction: a map $f$ has a dagger-Drazin inverse exactly when the positive endomorphisms $f f^\dagger$ and $f^\dagger f$ have ordinary Drazin inverses, and then the inverse is $f^\partial = f^\dagger (f f^\dagger)^D = (f^\dagger f)^D f^\dagger$. A second theorem shows a map has a Moore-Penrose inverse exactly when it is itself a dagger-Drazin inverse, placing Moore-Penrose inverses as the index-one case of the new concept. If correct, this unifies the categorical theory of Drazin inverses with Moore-Penrose inverses and gives a uniform existence criterion in terms of positivity.

What carries the argument

The load-bearing object is the dagger-Drazin inverse $f^\partial: B \to A$ of an arbitrary map $f: A \to B$ in a dagger category. The mechanism is a two-way transfer between $f$ and its positive endomorphisms $f f^\dagger$ and $f^\dagger f$. One direction builds the Drazin inverse of $f f^\dagger$ out of $f^\partial$ by setting $(f f^\dagger)^D = f^{\partial\dagger} f^\partial$; the other rebuilds $f^\partial$ from the Drazin inverse of $f f^\dagger$ by setting $f^\partial = f^\dagger (f f^\dagger)^D$. The transfer is powered by two lemmas from the authors' earlier categorical Drazin paper: the dagger preserves Drazin invertibility with $(x^\dagger)^D = (x^D)^\dagger$, and the product identity $(g h)^D g = g (h g)^D$ holds. These lemmas make the two directions of Theorem 3.6 go through and give the closed-form formulas.

What would settle it

Work in the category of complex matrices with the conjugate-transpose dagger and search randomly over matrices $f$: compute $f f^*$, obtain $(f f^*)^D$ with a Drazin-inverse routine, put $f^\partial = f^* (f f^*)^D$, and check the four dagger-Drazin axioms. The paper predicts the axioms always hold, and that $f^\partial$ agrees with the Moore-Penrose inverse when the Drazin index of $f f^*$ is at most one; any matrix where the axioms fail, or where an index-one $f^\partial$ differs from the SVD-based Moore-Penrose inverse, would refute Theorems 3.6 and 4.2.

Watch

Extended reading notes

Core claim

The central claim is that the dagger-Drazin inverse is the correct generalization of the Drazin inverse from endomorphisms to arbitrary maps in a dagger category. For $f: A \to B$, a dagger-Drazin inverse is a map $f^\partial: B \to A$ satisfying four axioms: some power of $f f^\dagger$ is fixed by $f f^\partial$ (and symmetrically for $f^\dagger f$ and $f^\partial f$), $f^\partial f f^\partial = f^\partial$, and both $f f^\partial$ and $f^\partial f$ are self-adjoint. The paper proves this inverse is unique and that it sits between the two classical generalized inverses. Theorem 3.6 states that $f$ has a dagger-Drazin inverse if and only if $f f^\dagger$ (equivalently $f^\dagger f$) has a Drazin inverse, with $f^\partial = f^\dagger (f f^\dagger)^D = (f^\dagger f)^D f^\dagger$; Theorem 4.2 states that $f$ has a Moore-Penrose inverse if and only if $f$ is itself a dagger-Drazin inverse (i.e., $f = g^\partial$ for some $g$), equivalently $f$ is dagger-Drazin and $f^{\partial\partial} = f$. The paper also proves at the level of categories that a dagger category is dagger-Drazin exactly when every positive map is Drazin, and that every ordinary Drazin dagger category is automatically dagger-Drazin.

Load-bearing premise

The equivalence between dagger-Drazin invertibility and Drazin invertibility of positive maps rests on two technical lemmas quoted from the authors' earlier paper—that $(x^\dagger)^D = (x^D)^\dagger$ and that $(gh)^D g = g(hg)^D$—and if either quoted lemma is false or misapplied, the main theorems do not follow from this paper alone.

Editorial extensions

If this is right

  • To decide whether $f$ has a dagger-Drazin inverse, it is enough to test one of $f f^\dagger$ or $f^\dagger f$ for an ordinary Drazin inverse; the dagger-Drazin index is the maximum of the two Drazin indices.
  • Every Drazin dagger category is dagger-Drazin, so in a category with a dagger where all endomorphisms have Drazin inverses—such as matrices over a field with transpose or conjugate transpose—every arrow of any type has a dagger-Drazin inverse.
  • Moore-Penrose invertibility coincides with being a dagger-Drazin inverse, so every Moore-Penrose inverse is the dagger-Drazin inverse of its own inverse, and it has dagger-Drazin index at most one.
  • Drazin inverses of opposing pairs in a plain category are exactly dagger-Drazin inverses of the corresponding maps in the cofree dagger category, with the same index.
  • For bounded linear operators between Hilbert spaces, dagger-Drazin invertibility of $f$ is equivalent to $f f^\dagger$ having finite ascent and descent, giving a concrete analytic criterion.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper does not draw out that the closed form $f^\partial = f^\dagger (f f^\dagger)^D$ gives a uniform numerical recipe: compute the Drazin inverse of the Gram operator $f f^\dagger$ once, then multiply on the left by $f^\dagger$; for index-one matrices this recipe is exactly the Moore-Penrose inverse.
  • The opposing-pair correspondence opens a dictionary between two theories: since cofree dagger categories are built from arbitrary categories, theorems about dagger-Drazin inverses automatically specialize to Drazin inverses of opposing pairs, a direction the paper leaves mostly implicit.
  • A testable extension would be to $\ast$-semigroups and $\ast$-rings: define $a^\partial = a^* (a a^*)^D$ whenever $a a^*$ is Drazin; the categorical proof structure suggests the same axioms should hold in any $\ast$-monoid with the analogous lemmas.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 8 minor

Summary. The paper introduces dagger-Drazin inverses for arbitrary maps in dagger categories, extending the classical notion of Drazin inverses for endomorphisms. Section 2 defines the inverse, proves uniqueness, and establishes basic properties (Proposition 2.6, Lemma 2.5). Section 3 proves the central equivalence (Theorem 3.6): a map f is dagger-Drazin invertible if and only if the induced positive maps ff† or f†f are Drazin, and gives closed forms for f^∂ and an index formula. Section 4 shows that Moore-Penrose invertibility is equivalent to being a dagger-Drazin inverse (Theorem 4.2). Section 5 relates Drazin inverses of opposing pairs to dagger-Drazin inverses in cofree dagger categories (Theorem 5.4). Examples are provided for matrices over involutive fields, bounded Hilbert-space operators, and partial injections.

Significance. If the main results are correct, this is a clean and natural categorical treatment of a broadly useful generalized inverse. The equivalence with Drazin invertibility of ff† and f†f (Theorem 3.6) is a strong structural statement, and the characterization of Moore-Penrose inverses as index-1 dagger-Drazin inverses (Theorem 4.2) adds a new perspective on a classical notion. The paper is largely self-contained, with proofs by explicit diagram chases; the only external dependencies are a few lemmas from the authors' earlier published work [9], which are clearly cited. The examples show the concept applies beyond matrices, including to Hilbert spaces and inverse categories. I found no fatal gap in the central derivations; the issues that remain are local and fixable.

minor comments (8)
  1. [Theorem 3.6, proof of (vi)] The quotation of [9, Lemma 7.5] is printed with a typo: the second equality appears as "h(gh)^D = (hg)hD" and should be "h(gh)^D = (hg)^D h". This identity is used to obtain the symmetric closed form f^∂ = (f†f)^D f†, so please correct the display and state the lemma's hypotheses explicitly (they are satisfied here because f being †-Drazin implies both ff† and f†f are Drazin).
  2. [Theorem 3.6] The enumerated list of consequences jumps from (iv) to (vi); there is no item (v). Please renumber (vi)–(viii) as (v)–(vii) or add a missing item.
  3. [Theorem 4.2, proof of (i)⇒(iii)] The argument verifies [D†.5] with j=1 but does not explicitly verify [D†.7] with m=1 (equivalently [G†.1.d]), so the asserted bound ind∂(f) ≤ 1 is not demonstrated in the text. The bound is true and follows from the Moore-Penrose axioms, for example via f† = f† f f∘, but a short derivation should be added.
  4. [Theorem 4.2, proof of (i)⇒(iii)] The correspondence of axioms is misstated: the correct pairings are [D†.2]↔[MP.2], [D†.3]↔[MP.3], and [D†.4]↔[MP.4].
  5. [Abstract] The phrase "a map has a Moore-Penrose inverse if and only if it is a Drazin inverse" should say "if and only if it is a dagger-Drazin inverse" (or "†-Drazin inverse"), to match Theorem 4.2.
  6. [Proposition 2.2, display (1)] The displayed equation appears to merge two separate identities for f^∂ and f^δ into one line, which is visually confusing; please split it into two equations.
  7. [Lemma 2.5] All proofs are left as exercises; while these checks are straightforward, a brief indication for part (iii) (partial isometries) and part (v) (†-idempotents) would improve self-containment.
  8. [Theorem 3.6(viii)] The index formula is asserted to be "straightforward to check"; since it is used in Corollary 3.7 and in Examples 3.10–3.14, please include a two-sentence proof, using the definitions of f^∂ and the Drazin axioms for ff† and f†f.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central equivalences are proved from the new definition, and the imported lemmas from the authors' prior work are independent categorical facts, not the target result.

full rationale

The paper's load-bearing results are Theorem 3.6 (f is dagger-Drazin iff ff† is Drazin) and Theorem 4.2 (f is Moore-Penrose iff f is a dagger-Drazin inverse). Both are argued by direct construction from the definitions rather than by assuming the conclusion. In Theorem 3.6, the forward direction builds (ff†)^D := f∂†f∂ and verifies the Drazin axioms; the reverse direction builds f∂ := f†(ff†)^D and verifies the dagger-Drazin axioms. The only externally imported ingredients are [9, Lemma 7.3] (gh Drazin iff hg Drazin), [9, Lemma 7.22] (dagger commutes with the Drazin inverse), and [9, Lemma 7.5] ((gh)^D g = g(hg)^D). These come from the authors' own earlier paper, but they are published, parameter-free categorical lemmas about ordinary Drazin inverses, stated independently of dagger-Drazin invertibility, and they do not presuppose the equivalence being proved. Under the stated review rules, such citations count as independent support rather than circularity. There is a real self-containment concern: Lemma 7.5 is quoted in Theorem 3.6 with an apparent typo, printed as '(hg)hD' instead of '(hg)^D h', and the paper does not re-prove it, so the closed form f∂ = f†(ff†)^D = (f†f)^D f† is contingent on the precise published statement. But that is a verification-gap issue, not a circular one, since the identity is not the paper's conclusion and is a standard consequence of Cline's formula. Theorem 4.2 also proceeds by checking the four Moore-Penrose axioms against the dagger-Drazin axioms, not by renaming. No fitted parameter is presented as a prediction, and no uniqueness theorem is invoked to force a choice. The paper's central content is therefore self-contained in the relevant sense, and the low score reflects only the minor reliance on prior work.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

No free parameters or empirical postulates appear in this pure-mathematics paper. The central claim rests on the dagger-category framework and on two published lemmas from the authors' earlier work, treated here as background theorems.

assumptions (4)
  • domain assumption Dagger-category axioms: each map f has an adjoint f† with (f†)† = f, (fg)† = g† f†, 1† = 1.
    All dagger-Drazin definitions and theorems are internal to a dagger category; without a global involution on maps the central object cannot be stated.
  • domain assumption Lemma 7.22 of [9]: in a dagger category, x is Drazin if and only if x† is Drazin, and then (x†)^D = (x^D)†.
    Used in Theorem 3.6 to show the Drazin inverse of f f† is self-adjoint, which is essential for verifying [D†.3] and [D†.4].
  • domain assumption Lemma 7.5 of [9]: for maps g,h with gh and hg Drazin, (gh)^D g = g (hg)^D.
    Used in Theorem 3.6(vi) to derive the two equivalent expressions for f^∂.
  • standard math Every square matrix over a field has a Drazin inverse (Campbell-Meyer [4] and [9, Sec 3.2]).
    Used in Examples 3.10-3.11 and 4.4 to conclude every matrix map is dagger-Drazin.
invented entities (1)
  • Dagger-Drazin inverse (f^∂)
    purpose: A new generalized inverse for arbitrary maps in a dagger category, satisfying [D†.1]-[D†.4].
    This is a new mathematical definition, not an empirical entity. It has no falsifiable handle outside the theory; its value is structural.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Dagger-Drazin Inverses." pith.science (2026). https://pith.science/paper/CSEODKRR

@misc{pith2026250205306,
  author       = {Pith},
  title        = {Pith review of: Dagger-Drazin Inverses},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CSEODKRR}},
  note         = {Machine review of arXiv:2502.05306}
}
read the original abstract

Drazin inverses are a special kind of generalized inverses that can be defined for endomorphisms in any category. A natural question to ask is whether one can somehow extend the notion of Drazin inverse to arbitrary maps - not simply endomorphisms. It turns out that this is possible and, indeed, natural to do so for dagger categories. This paper, thus, introduces the notion of a dagger-Drazin inverse, which is a new kind of generalized inverse appropriate for arbitrary maps in a dagger category. This inverse is closely related to the Drazin inverse, for having dagger-Drazin inverses is equivalent to asking that positive maps have Drazin inverses. Moreover, dagger-Drazin inverses are also closely related to Moore-Penrose inverses as we observe that a map has a Moore-Penrose inverse if and only if it is a Drazin inverse. Furthermore, we explain how Drazin inverses of opposing pairs correspond precisely to dagger-Drazin inverses in cofree dagger categories. We also give examples of dagger-Drazin inverses for matrices over (involutive) fields, bounded linear operators, and partial injections.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 25 canonical work pages

  1. [9]

    Cockett, J.-S

    R. Cockett, J.-S. P . Lemay & P . V . Srinivasan (2024): Drazin Inverses in Categories . Theory and Applica- tions of Categories 43(14), pp. 455–519. Available at http://www.tac.mta.ca/tac/volumes/43/14/ 43-14abs.html

    work page 2024

  2. [1]

    Azumaya (1954): Strongly π -regular rings

    G. Azumaya (1954): Strongly π -regular rings. Journal of the Faculty of Science Hokkaido University. Ser. 1 Mathematics 13(1), pp. 034–039. Available at http://hdl.handle.net/2115/55983

    work page 1954

  3. [2]

    O. M. Baksalary & G. Trenkler (2021): The Moore–Penrose inverse: a hundred years on a frontline of physics research. The European Physical Journal H 46, pp. 1–10, doi: 10.1140/epjh/s13129-021-00011-y

    work page doi:10.1140/epjh/s13129-021-00011-y 2021

  4. [3]

    Bartha (2014): Quantum Turing automata

    M. Bartha (2014): Quantum Turing automata . In Benedikt L¨ owe & Glynn Winskel, editors: Proceedings 8th International Workshop on Developments in Computational Models, Cambridge, United Kingdom, 17 June 2012, Electronic Proceedings in Theoretical Computer Science 143, Open Publishing Association, pp. 17–31, doi: 10.4204/EPTCS.143.2

    work page doi:10.4204/eptcs.143.2 2014

  5. [4]

    Campbell & C

    S. Campbell & C. Meyer (2009): Generalized inverses of linear transformations . SIAM, doi: 10.1137/1. 9780898719048

    doi:10.1137/1 2009

  6. [5]

    N. Cao, J. Lin, D. Kribs, Y .-T. Poon, B. Zeng & R. Laflamme (2 021): NISQ: Error correction, mitigation, and noise simulation . arXiv preprint arXiv:2111.02345

    work page Pith review arXiv

  7. [6]

    J. Chen, Y . Wang, J. Chen & S. Su (2023): Optimal figure of merit of low-dissipation quantum refriger ators. Physical Review E 107(4), p. 044118, doi: 10.1103/PhysRevE.107.044118

    work page doi:10.1103/physreve.107.044118 2023

  8. [7]

    J.-F. Chen, C. Sun & H. Dong (2021): Extrapolating the thermodynamic length with finite-time me asure- ments. Physical Review E 104(3), p. 034117, doi: 10.1103/PhysRevE.104.034117

    work page doi:10.1103/physreve.104.034117 2021

Show all 30 references
  1. [8]

    Cockett & J-S

    J.R.B. Cockett & J-S. P . Lemay (2023): Moore-Penrose Dagger Categories. In: Proceedings of the Twentieth International Conference on Quantum Physics and Logic, Paris, France, 17-21st July 2023, Electronic Pro- ceedings in Theoretical Computer Science 384, Open Publishing Asso...

    2023

  2. [10]

    M. P . Drazin (1958): Pseudo-inverses in associative rings and semigroups . The American mathematical monthly 65(7), pp. 506–514, doi: 10.2307/2308576

    1958 doi

  3. [11]

    Fitting (1933): Die Theorie der Automorphismenringe Abelscher Gruppen und ihr Analogon bei nicht kommutativen Gruppen

    H. Fitting (1933): Die Theorie der Automorphismenringe Abelscher Gruppen und ihr Analogon bei nicht kommutativen Gruppen. Mathematische Annalen 107, pp. 514–542, doi: 10.1007/BF01448909

    1933 doi

  4. [12]

    Heunen (2009): Categorical quantum models and logics

    C. Heunen (2009): Categorical quantum models and logics . Amsterdam University Press, doi: 10.5117/ 9789085550242

    2009

  5. [13]

    Heunen & J

    C. Heunen & J. Vicary (2019): Categories for Quantum Theory: an Introduction . Oxford University Press, doi:10.1093/oso/9780198739623.001.0001

    2019

  6. [14]

    Jacobs, C

    B. Jacobs, C. Heunen & I. Hasuo (2009): Categorical semantics for arrows. Journal of functional program- ming 19(3-4), pp. 403–438, doi: 10.1017/S0956796809007308

    2009 doi

  7. [15]

    C. F. King (1977): A note on Drazin inverses . Pacific Journal of Mathematics 70, pp. 383–390, doi: 10. 2140/pjm.1977.70.383

    1977

  8. [16]

    Leinster (2023): The eventual image

    T. Leinster (2023): The eventual image. Theory and Applications of Categories . Available at http://www. tac.mta.ca/tac/volumes/42/9/42-09abs.html

    2023

  9. [17]

    Miller, M

    H. Miller, M. Scandi, J. Anders & M. Perarnau-Llobet (20 19): W ork fluctuations in slow processes: quantum signatures and optimal control . Physical review letters 123(23), p. 230603, doi: 10.1103/PhysRevLett. 123.230603

    doi

  10. [18]

    E. H. Moore (1920): On the reciprocal of the general algebraic matrix . Bull. Am. Math. Soc. 26, pp. 394– 395, doi:10.1090/S0002-9904-1920-03322-7

    1920 doi

  11. [19]

    W . D. Munn (1961): Pseudo-inverses in semigroups. In: Mathematical Proceedings of the Cambridge Philo- sophical Society, 57, Cambridge University Press, pp. 247–250, doi: 10.1017/S0305004100035143. 316 Dagger-Drazin Inverses

    1961 doi

  12. [20]

    Penrose (1955): A generalized inverse for matrices

    R. Penrose (1955): A generalized inverse for matrices . In: Mathematical proceedings of the Cambridge philosophical society, 51, Cambridge University Press, pp. 406–413, doi: 10.1017/S0305004100030401

    1955 doi

  13. [21]

    Puystjens & M.C

    R. Puystjens & M.C. Gouveia (2004): Drazin invertibility for matrices over an arbitrary ring . Linear algebra and its applications 385, pp. 105–116, doi: 10.1016/S0024-3795(03)00618-9

    2004 doi

  14. [22]

    Puystjens & D

    R. Puystjens & D. w. Robinson (1981): The Moore-Penrose inverse of a morphism with factorization. Linear Algebra and its Applications 40, pp. 129–141, doi: 10.1016/0024-3795(81)90145-2

    1981 doi

  15. [23]

    Puystjens & D

    R. Puystjens & D. W . Robinson (1984): The Moore-Penrose inverse of a morphism in an additive categ ory. Communications in algebra 12(3), pp. 287–299, doi: 10.1080/00927878408823004

    1984 doi

  16. [24]

    Puystjens & D

    R. Puystjens & D. W . Robinson (1985): EP morphisms. Linear algebra and its applications 64, pp. 157–174, doi:10.1016/0024-3795(85)90273-3

    1985 doi

  17. [25]

    Puystjens & D

    R. Puystjens & D. W . Robinson (1987): Generalized inverses of morphisms with kernels. Linear Algebra and Its Applications 96, pp. 65–86, doi: 10.1016/0024-3795(87)90336-3

    1987 doi

  18. [26]

    Puystjens & D

    R. Puystjens & D. W . Robinson (1990): Symmetric morphisms and the existence of Moore-Penrose inv erses. Linear Algebra and Its Applications 131, pp. 51–69, doi: 10.1016/0024-3795(90)90374-L

    1990 doi

  19. [27]

    Selinger (2007): Dagger compact closed categories and completely positive m aps

    P . Selinger (2007): Dagger compact closed categories and completely positive m aps. Electronic Notes in Theoretical computer science 170, pp. 139–163, doi: 10.1016/j.entcs.2006.12.018

    2007 doi

  20. [28]

    Wang (2017): A class of Drazin inverses in rings

    Z. Wang (2017): A class of Drazin inverses in rings . Filomat 31(6), pp. 1781–1789, doi: 10.2298/ FIL1706781W

    2017

  21. [29]

    Y . Wei & S. Qiao (2003): The representation and approximation of the Drazin inverse of a linear operator in Hilbert space . Applied Mathematics and Computation 138(1), pp. 77–89, doi: 10.1016/ S0096-3003(02)00100-5

    2003

  22. [30]

    Z. Wu, S. Zhang & C. Zhu (2014): Remarks on generalized quantum gates . Hacettepe Journal of Math- ematics and Statistics 43(3), pp. 451–460. Available at https://dergipark.org.tr/tr/download/ article-file/649195

    2014

Pith tools

Reviewed August 8, 2026 · model on record in the stance chip above.