Pith. sign in

REVIEW 1 major objections 5 minor 32 references

A Newton–Schulz iteration, initialized by a scaled i-conjugate transpose, converges to the Moore–Penrose inverse of any rectangular or rank-deficient split-quaternion matrix, with exact residual formulas certifying the convergence rate.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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

T0 review · deepseek-v4-flash

2026-08-03 10:18 UTC pith:HK3VPI6A

load-bearing objection Main theorems are sound: the thin-SVD Newton–Schulz analysis is clean and the split-quaternion transfer works, but there is a typo in the scalar Moore–Penrose formula and the experimental reproducibility is not as claimed. the 1 major comments →

arxiv 2607.29270 v1 pith:HK3VPI6A submitted 2026-07-31 math.NA cs.NA

Iterative Methods for Computing the Moore--Penrose Inverse of Split-Quaternion Matrices with Applications

Salman Ahmadi-Asl , Valentin Leplat , Mohammad S. Alkousa This is my paper
classification math.NA cs.NA MSC 15A0915A6665F1065F35
keywords split-quaternion matrixMoore–Penrose inverseNewton–Schulz iterationpolynomial initializationreal representationCUR approximationzero divisorsi-conjugate transpose
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper establishes that the Moore–Penrose inverse of a split-quaternion matrix can be computed by a Newton–Schulz iteration, with convergence guaranteed even for rectangular, rank-deficient, and zero-divisor cases. Using a 2×2 real representation and the i-conjugate transpose, it derives exact formulas for the iterates and projector residuals, and shows that an embedded real implementation and a native split-quaternion implementation are exactly equivalent. It also proves that for fixed sampled rows and columns, the CUR middle factor C†AR† is Frobenius-optimal, and gives a rank condition under which the cheaper cross factor W† yields exact reconstruction. This matters because split quaternions contain zero divisors and have an indefinite norm, so generalized inverses are non-obvious; certified iterative algorithms with explicit iteration counts make such computations practical for large or low-rank problems.

Core claim

Starting from a real matrix M of rank r with positive singular values σ1≥…≥σr, the paper proves that the Newton–Schulz iteration X_{k+1}=2X_k−X_kMX_k with X_0=αM^T converges to M† whenever 0<α<2/σ1^2. The proof uses a thin SVD to obtain exact formulas: X_k=V_r diag((1−θ_i^{2^k})/σ_i)U_r^T and the projector residual PL−MX_k=U_r diag(θ_i^{2^k})U_r^T, with θ_i=1−ασ_i^2. These formulas are transferred to split-quaternion matrices through the real representation σ and the i-conjugate transpose H, for which (A^H)_σ=(A_σ)^T. Both the embedded real iteration and the native split-quaternion iteration are shown to converge to A†, and the CUR middle factor C†AR† is shown to be Frobenius-optimal for fix

What carries the argument

The machinery is the block 2×2 real representation σ (from split quaternions to real 2×2 matrices) and its matrix extension, together with the i-conjugate transpose H satisfying (A^H)_σ=(A_σ)^T. The real representation converts split-quaternion arithmetic into real matrix arithmetic in a product-preserving way (Frobenius norms scale by √2), so the thin SVD of A_σ reduces the iteration to independent scalar recurrences per singular direction. The H-adjoint is exactly what makes the real transpose correspond to the split-quaternion adjoint, so the four Penrose equations carry over unchanged. Each singular-direction coefficient d_{i,k} follows the scalar Newton map d←d(2−σ_i d), whose error squ

Load-bearing premise

The load-bearing premise is that the Moore–Penrose inverse is defined with respect to the i-conjugate transpose, because only then does the real transpose (A_σ)^T correspond to the adjoint used in the Penrose equations; if the full conjugate is the intended adjoint, every algorithm in the paper converges to a different object.

What would settle it

Take the 2×2 real matrix M=diag(1,0) and run X_{k+1}=2X_k−X_kMX_k with X_0=αM^T. For α=1.5 the theorem predicts X_1=diag(0.75,0), X_2=diag(0.9375,0), converging to diag(1,0); for α=2.0 it predicts non-convergence (the (1,1) entry alternates between 0 and 2). If either prediction fails, the thin-SVD convergence claim is wrong. For the split-quaternion side, computing q† for the zero divisor q=1+j via the native iteration and comparing with τ(q)/(2∥q∥_E^2) would test the transfer.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • For every split-quaternion matrix A, the native iteration X_{k+1}=2X_k−X_k A X_k with X_0=αA^H converges to A† whenever 0<α<2/∥Aσ∥_2^2, and it is exactly equivalent to the embedded real iteration in exact arithmetic.
  • The normalized residual ∥A X_k A−A∥_F / ∥A∥_F is a certified stopping criterion: Theorem 4.4 gives ∥X_k−A†∥_F ≤ ∥A†∥_2^2 ∥A X_k A−A∥_F, so users can stop on a computable quantity without knowing the exact inverse.
  • For fixed sampled column and row sets, C†AR† minimizes ∥A−CUR∥_F over all k×k split-quaternion middle factors U, and when rankσ(A(R,C))=rankσ(A), the cross factor W† equals C†AR† and reconstructs A exactly.
  • The LS–Gram polynomial initialization, supplemented by a spectral acceptance test and a safe scaled-transpose fallback, never sacrifices the convergence guarantee and reduces the number of Newton–Schulz iterations in the reported tests (about 20% fewer than a scaled transpose at κ=10^3).

Where Pith is reading between the lines

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

  • Because the Moore–Penrose inverse here is defined with respect to the i-conjugate transpose, the same iteration applied with the full conjugate transpose would compute a different object; applications that use the full conjugate (e.g., some Lorentzian formulations) should translate the scaling and residual formulas before adopting the algorithm.
  • The exact residual–error identity suggests a direct a posteriori error bound that could be used for adaptive stopping or as a certificate in verified numerical libraries, a use the paper does not develop.
  • The same real-representation transfer should apply to other generalized inverses (e.g., the Drazin inverse) and to other hypercomplex algebras with a fixed anti-involution, since the paper's proof only needs (A^H)_σ=(A_σ)^T and the real Newton–Schulz theorem.
  • The polynomial initialization's benefits are shown for dense matrices; a testable extension is to use it in matrix-free or sparse settings where Gram-matrix powers are approximated, since the setup cost currently dominates for small matrices.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 5 minor

Summary. The paper develops iterative Newton–Schulz algorithms for the Moore–Penrose inverse of split-quaternion matrices via a 2×2 real representation and the i-conjugate transpose H, so that (A^H)_σ=(A_σ)^T. It defines A† by σ^{-1}((Aσ)†), proves scalar formulas including zero-divisor cases, and establishes existence and uniqueness of the H-based Moore–Penrose inverse. The main analytic result, Theorem 4.1, gives exact thin-SVD expressions for the iterates and residuals of the Newton–Schulz iteration for arbitrary-rank real rectangular matrices. These results are transferred to embedded and native split-quaternion iterations in Theorems 4.6 and 4.7. A Souriau-inspired low-degree polynomial initialization with a spectral acceptance test and safe fallback is proposed in Section 4.6. The paper then proves that U⋆=C†AR† is a Frobenius-optimal CUR middle factor and gives a rank condition under which the cross factor W† is exact. Numerical examples in MATLAB and Python illustrate the behavior, including an ablation of the polynomial initialization.

Significance. If the results stand, this is the first iteration-count-certified Newton–Schulz analysis for split-quaternion Moore–Penrose inverses without full-rank restrictions. The thin-SVD argument in Theorem 4.1 is clean and yields exact formulas for iterates and residuals rather than asymptotic rates, and the transfer through the real representation is elegant. The CUR/cross optimality results are useful and are proved without restrictive rank assumptions. I agree with the stress-test assessment that the main derivations are sound and non-circular: the Moore–Penrose inverse is defined through the real representation and then shown to satisfy the Penrose equations, and the convergence theorems are derived from a thin SVD without fitted constants. The paper is also careful to label the timing examples as single-realization implementation checks and to disclose that the polynomial-initialization experiments use exactly known spectra and ρ. The principal caveat is that the object computed is the H-adjoint Moore–Penrose inverse, not the full-conjugate inverse; the paper is explicit about this, but it should be kept prominent.

major comments (1)
  1. [§3.1, Eq. (17)] The first branch of the scalar formula is stated as q†=q/N(q) when N(q)≠0. This is false: for q=i it gives q†=i, yet i·i·i=-i≠i. The proof in the same proposition derives Q^{-1}=(\bar{q})_σ/N(q), which corresponds to q^{-1}=\bar{q}/N(q); the displayed formula is missing the full conjugate. This is a must-fix. It does not propagate to Theorem 4.1, 4.7, 5.1, or 5.3, which use Definition 3.3, but the scalar formula is part of the paper's stated framework and should be corrected.
minor comments (5)
  1. [§6.4] The text says the complete Python implementation is available in a Google Colab notebook, but no URL or repository is given. Please provide a link or attach the code as supplementary material.
  2. [§4.6, Algorithm 4] The LS–Gram initialization is used only when full row/column rank is certified, and the iteration requires a certified upper bound ρ≥||M||_2^2. No certification procedure is specified. Please state how rank certification and the spectral-norm bound are obtained in practice (e.g., rank-revealing QR, power iteration/Lanczos with a safety factor) and how an overestimate of ρ affects the polynomial fit.
  3. [§2.2 / Abstract] The Moore–Penrose inverse is defined with respect to the i-conjugate transpose H, not the full conjugate A^*. Although Section 2.2 is explicit, the abstract and title could mislead readers familiar with other split-quaternion conventions. Consider stating this qualifier in the abstract.
  4. [§6.1–6.3] The timing comparisons are based on one matrix realization each. The paper labels them as implementation checks, which is appropriate; if they are meant to support performance claims, multiple seeds and interquartile ranges should be reported as in Example 6.4.
  5. [§4.5, Prop. 4.5 proof] Minor typo: 'Sigma' appears instead of '\Sigma' in the thin-SVD expression for the Frobenius norm of αM^T−M†. Also, check the hyphenation of 'split-quaternion' for consistency.

Circularity Check

0 steps flagged

No significant circularity; the derivation is self-contained.

full rationale

No circular step is present. The Moore–Penrose inverse is deliberately defined through the real representation in Definition 3.3, and Theorem 3.4 independently proves that this object satisfies and is uniquely determined by the four Penrose equations, using the standard real Penrose theory. Theorem 4.1 derives exact Newton–Schulz convergence for arbitrary-rank real matrices from a thin SVD, with no fitted constants; the split-quaternion versions (Theorems 4.6 and 4.7) follow by the representation isomorphism and do not depend on any unverified prior result. The LS–Gram polynomial is a heuristic initialization fitted to the Gram spectrum, but Theorem 4.9's convergence guarantee is conditional on the independently checked spectral residual η_d<1, and the numerical experiments are implementation tests rather than predictions forced by the fit. Self-citations [24, 25] are contextual or used as comparison/background, not as load-bearing inputs to the main proofs. Even the noted Proposition 3.1 branch appears to contain a bar-q/N(q) typo, a statement/proof mismatch, not a circularity. The choice of the i-conjugate transpose is an explicit convention whose consequences are proven, not assumed into the results.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The central proofs are self-contained given standard real linear algebra. The only external result invoked for an alternative characterization is the split-quaternion SVD of [24]. Algorithmic hyperparameters (degree d, safety δ, scaling β) and LS–Gram coefficients are the tunable/fitted quantities; no new scientific entities are introduced.

free parameters (4)
  • polynomial degree d = 4 (chosen by ablation in Table 4)
    Controls the LS–Gram initialization cost vs. iteration savings; the paper tests d=0,2,4,6 and picks d=4.
  • safety margin δ = not specified
    Algorithm 4 accepts the polynomial initialization only when estimated η < 1−δ; no numerical value is given in the manuscript, and the experiments only report that all initializations passed.
  • fallback scaling β = β∈(0,2); experiments use 0.9 (Example 6.1), 1 elsewhere
    Initial scale α=β/ρ must lie in the convergence interval; the exact value is an implementation choice, not derived.
  • LS–Gram coefficients c0..cd = least-squares solution of (52)
    These are fitted to the normalized Gram spectrum of the specific matrix; they are algorithmic outputs, not independent parameters, but they are 'fitted to data' in the literal sense.
axioms (4)
  • standard math The real Moore–Penrose inverse exists, is unique, and satisfies the four Penrose equations, together with the thin SVD of any real matrix.
    Used throughout Section 4 and Theorem 3.4; standard background.
  • standard math The representation σ and its block extension are algebra isomorphisms and (A^H)_σ=(Aσ)^T.
    Proven in Lemmas 2.1–2.2 and used as the foundation of the paper.
  • domain assumption Existence and form of the split-quaternion SVD (Theorem 3.6) as given in [24].
    Invoked in Remark 3.7 as an alternative characterization; not used in the convergence proofs or algorithms.
  • standard math The reverse-order law (XY)†=Y†X† when X has full column rank and Y has full row rank.
    Used in Theorem 5.3; proven within the paper via full-rank formulas.

pith-pipeline@v1.3.0-daily-deepseek · 21369 in / 25738 out tokens · 237069 ms · 2026-08-03T10:18:54.881398+00:00 · methodology

0 comments
read the original abstract

We study iterative methods for computing the Moore--Penrose inverse of split-quaternion matrices. We first establish a consistent framework based on a \(2\times2\) real representation and the associated \(i\)-conjugate transpose. This representation gives a direct definition of the Moore--Penrose inverse and clarifies the treatment of nonzero zero divisors. We then analyze Newton--Schulz iterations for rectangular and rank-deficient matrices. Using a thin singular value decomposition of the real representative, we derive the convergence conditions, the evolution of the projector residuals, and an exact relation between the residual and the error. The resulting analysis applies both to an embedded real implementation and to a native split-quaternion iteration, which are shown to be equivalent under the representation. We also propose a low-degree polynomial initialization inspired by Souriau's inverse recursion. The polynomial is obtained by a least-squares approximation of the inverse Gram factor and is combined with a spectral acceptance test and a safe fallback initialization. Finally, we apply the proposed methods to cross and CUR approximations of split-quaternion matrices. We characterize the optimal middle factor for fixed sampled rows and columns and give conditions under which the cheaper cross factor gives an exact reconstruction. Numerical experiments illustrate the accuracy of the methods and the practical benefit of the polynomial initialization.

Figures

Figures reproduced from arXiv: 2607.29270 by Mohammad S. Alkousa, Salman Ahmadi-Asl, Valentin Leplat.

Figure 1
Figure 1. Figure 1: Total time as the matrix size increases, with [PITH_FULL_IMAGE:figures/full_fig_p032_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Newton–Schulz iterations as the prescribed condition number increases for split [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗

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

32 extracted references · 5 canonical work pages

  1. [1]

    Ahmadi-Asl, F

    S. Ahmadi-Asl, F. P. A. Beik, Iterative algorithms for least-squares solutions of a quater- nion matrix equation, Journal of Applied Mathematics and Computing 53 (1) (2017) 95– 127

    2017

  2. [2]

    Ahmadi-Asl, F

    S. Ahmadi-Asl, F. P. A. Beik, An efficient iterative algorithm for quaternionic least-squares problems over the generalized-(anti-) bi-hermitian matrices, Linear and Multilinear Alge- bra 65 (9) (2017) 1743–1769

    2017

  3. [3]

    I. I. Kyrchei, Quaternion Generalized Inverses: Foundations, Theory, Problems, and Solu- tions, Academic Press, 2025

    2025

  4. [4]

    Le Bihan, J

    N. Le Bihan, J. Mars, Singular value decomposition of quaternion matrices: a new tool for vector-sensor signal processing, Signal processing 84 (7) (2004) 1177–1199

    2004

  5. [5]

    Ahmadi-Asl, M

    S. Ahmadi-Asl, M. Nobakht Kooshkghazi, V . Leplat, Pass-efficient randomized algo- rithms for low-rank approximation of quaternion matrices, Numerical Algorithms (2026). doi:10.1007/s11075-026-02362-3

    work page doi:10.1007/s11075-026-02362-3 2026

  6. [6]

    K. Shoemake, Animating rotation with quaternion curves, in: Proceedings of the 12th Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH ’85, Association for Computing Machinery, 1985, pp. 245–254.doi:10.1145/325334. 325242

    work page doi:10.1145/325334 1985

  7. [7]

    J. S.-C. Yuan, Closed-loop manipulator control using quaternion feedback, IEEE Journal on Robotics and Automation 4 (4) (1988) 434–440.doi:10.1109/56.809

    work page doi:10.1109/56.809 1988

  8. [8]

    Finkelstein, J

    D. Finkelstein, J. M. Jauch, S. Schiminovich, D. Speiser, Foundations of quaternion quantum mechanics, Journal of Mathematical Physics 3 (2) (1962) 207–220.doi: 10.1063/1.1703794

    work page doi:10.1063/1.1703794 1962

  9. [9]

    De Leo, G

    S. De Leo, G. Scolarici, Right eigenvalue equation in quaternionic quantum mechanics, Journal of Physics A: Mathematical and General 33 (15) (2000) 2971–2995

    2000

  10. [10]

    W. R. Hamilton, On a new species of imaginary quantities connected with a theory of quaternions, Proceedings of the Royal Irish Academy 2 (1844) 424–434

  11. [11]

    Cockle, Iii

    J. Cockle, Iii. on a new imaginary in algebra, The London, Edinburgh, and Dublin Philo- sophical Magazine and Journal of Science 34 (226) (1849) 37–47

  12. [12]

    Z. ¨Ozdemir, A kinematic model of the rytov’s law in the optical fiber via split quaternions: application to electromagnetic theory, The European Physical Journal Plus 137 (6) (2022) 651

    2022

  13. [13]

    Antonuccio, Split-quaternions and the dirac equation, Advances in Applied Clifford Algebras 25 (1) (2015) 13–29

    F. Antonuccio, Split-quaternions and the dirac equation, Advances in Applied Clifford Algebras 25 (1) (2015) 13–29

    2015

  14. [14]

    ¨Ozdemir, M

    M. ¨Ozdemir, M. Erdo˘gdu, H. S ¸ims ¸ek, On the eigenvalues and eigenvectors of a lorentzian rotation matrix by using split quaternions, Advances in Applied Clifford Algebras 24 (1) (2014) 179–192. 34

    2014

  15. [15]

    Wood, Quaternionic eigenvalues, Bulletin of the London Mathematical Society 17 (2) (1985) 137–138

    R. Wood, Quaternionic eigenvalues, Bulletin of the London Mathematical Society 17 (2) (1985) 137–138

    1985

  16. [16]

    ¨Ozdemir, The roots of a split quaternion, Applied mathematics letters 22 (2) (2009) 258–263

    M. ¨Ozdemir, The roots of a split quaternion, Applied mathematics letters 22 (2) (2009) 258–263

    2009

  17. [17]

    L. Kula, Y . Yayli, Split quaternions and rotations in semi euclidean space e 4 2, Journal of the Korean Mathematical Society 44 (6) (2007) 1313–1327

    2007

  18. [18]

    Alag¨oz, K

    Y . Alag¨oz, K. H. Oral, S. Y ¨uce, Split quaternion matrices, Miskolc Mathematical Notes 13 (2) (2012) 223–232

    2012

  19. [19]

    Si, Q.-W

    K.-W. Si, Q.-W. Wang, L.-M. Xie, A classical system of matrix equations over the split quaternion algebra, Advances in Applied Clifford Algebras 34 (5) (2024) 51

    2024

  20. [20]

    X. Liu, Y . Zhang, Consistency of split quaternion matrix equations ax⋆ −xb=cy+d and x−ax ⋆b=cy+d , Advances in Applied Clifford Algebras 29 (4) (2019) 64

    2019

  21. [21]

    Gao, Q.-W

    Z.-H. Gao, Q.-W. Wang, L.-m. Xie, The (anti-)η-hermitian solution to a novel system of matrix equations over the split quaternion algebra, Mathematical Methods in the Applied Sciences 47 (18) (2024) 13896–13913

    2024

  22. [22]

    Abłamowicz, The moore–penrose inverse and singular value decomposition of split quaternions, Advances in Applied Clifford Algebras 30 (3) (2020) 33

    R. Abłamowicz, The moore–penrose inverse and singular value decomposition of split quaternions, Advances in Applied Clifford Algebras 30 (3) (2020) 33

    2020

  23. [23]

    W. Cao, Z. Chang, The Moore–Penrose inverses of split quaternions, Linear and Multilin- ear Algebra 70 (9) (2022) 1631–1647.doi:10.1080/03081087.2020.1769015

    arXiv 2022

  24. [24]

    G. Wang, T. Jiang, V . Vasil’ev, Z. Guo, On singular value decomposition for split quater- nion matrices and applications in split quaternionic mechanics, Journal of Computational and Applied Mathematics 436 (2024) 115447

    2024

  25. [25]

    Leplat, S

    V . Leplat, S. Ahmadi-Asl, J. Pan, N. Zheng, Iterative methods for computing the Moore– Penrose pseudoinverse of quaternion matrices, with applications, Journal of Scientific Computing 107 (3) (2026) 86.doi:10.1007/s10915-026-03304-w. URLhttps://doi.org/10.1007/s10915-026-03304-w

    work page doi:10.1007/s10915-026-03304-w 2026

  26. [26]

    Schulz, Iterative berechung der reziproken matrix, ZAMM-Journal of Applied Math- ematics and Mechanics/Zeitschrift f ¨ur Angewandte Mathematik und Mechanik 13 (1) (1933) 57–59

    G. Schulz, Iterative berechung der reziproken matrix, ZAMM-Journal of Applied Math- ematics and Mechanics/Zeitschrift f ¨ur Angewandte Mathematik und Mechanik 13 (1) (1933) 57–59

    1933

  27. [27]

    Ben-Israel, T

    A. Ben-Israel, T. N. Greville, Generalized inverses: theory and applications, Springer, 2003

    2003

  28. [28]

    Souriau, Une m ´ethode nouvelle pour la d ´ecomposition spectrale et l’inversion des matrices, Acad

    J.-M. Souriau, Une m ´ethode nouvelle pour la d ´ecomposition spectrale et l’inversion des matrices, Acad. Sc., Paris (1948)

    1948

  29. [29]

    Souriau, Grammaire de la nature, 2007, version du 8 juillet 2007

    J.-M. Souriau, Grammaire de la nature, 2007, version du 8 juillet 2007. 35

    2007

  30. [30]

    S. A. Goreinov, N. L. Zamarashkin, E. E. Tyrtyshnikov, Pseudo-skeleton approximations by matrices of maximal volume, Mathematical Notes 62 (4) (1997) 515–519

    1997

  31. [31]

    Allen, M.-J

    K. Allen, M.-J. Lai, Z. Shen, Maximal volume matrix cross approximation for image compression and least squares solution, Advances in Computational Mathematics 50 (5) (2024) 102

    2024

  32. [32]

    Kliushev, J

    V . Kliushev, J. Pan, V . Leplat, Quaternion maximum-volume submatrix selection with applications to multichannel imaging and visual data (2026).arXiv:2606.08175. A Auxiliary proofs This appendix collects proofs of standard representation and Moore–Penrose identities used in the main text. A.1 Proof of Lemma 2.1 Proof.The additivity and real homogeneity f...

    Pith/arXiv arXiv 2026