On solutions of singular Sylvester equations in quaternions

Christian Scheunert; Frank H. P. Fitzek; Hristina Radak

arxiv: 2510.04629 · v7 · pith:QPHXACABnew · submitted 2025-10-06 · 🧮 math.RA

On solutions of singular Sylvester equations in quaternions

Hristina Radak , Christian Scheunert , Frank H. P. Fitzek This is my paper

Pith reviewed 2026-05-18 09:45 UTC · model grok-4.3

classification 🧮 math.RA

keywords quaternionsSylvester equationsingular equationsexistence conditionssquare rootslinear algebra over quaternions

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The pith

Solutions to the singular Sylvester equations ax - xb = 0 and ax - xb = c exist in quaternions under specific conditions and are constructed using quaternion square roots.

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

The paper examines the homogeneous and inhomogeneous Sylvester equations over the quaternions. It establishes conditions under which solutions exist and provides explicit expressions for the general solution as well as nonzero solutions. These derivations rely on taking square roots in the quaternion algebra. A reader might care because quaternions model rotations and other phenomena where non-commutativity plays a role, so solvability results here could inform practical computations in those domains.

Core claim

The authors show that for the equations ax - xb = 0 and ax - xb = c over quaternions, existence of solutions depends on certain relations involving a and b, and the solutions can be obtained by applying quaternion square roots to construct the general form.

What carries the argument

Quaternion square roots, used to derive the general and nonzero solutions in the singular case.

If this is right

If the existence conditions hold, the general solution can be parameterized explicitly.
Nonzero solutions are available when the square root construction succeeds.
The same approach applies to both the homogeneous and inhomogeneous versions of the equation.

Where Pith is reading between the lines

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

This construction may extend to finding solutions in other division algebras.
Numerical algorithms could be developed based on computing quaternion square roots for these equations.
Applications in engineering fields using quaternions, such as attitude control, might benefit from these closed-form solutions.

Load-bearing premise

That quaternion square roots can always be chosen so that the resulting expressions satisfy the original equation without additional restrictions on the coefficients.

What would settle it

Finding specific quaternions a, b, and c where no choice of square root yields a solution that satisfies ax - xb = c would falsify the general construction.

read the original abstract

The quaternionic equations ax-xb=0 and ax-xb=c are investigated, which are called homogeneous and inhomogeneous Sylvester equations, respectively. Conditions for the existence of solutions are provided. In addition, the general and nonzero solutions to these equations are derived applying quaternion square roots.

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.

Desk Editor's Note private letter to a colleague

Paper gives explicit existence conditions and square-root formulas for singular quaternionic Sylvester equations, but the construction may need extra checks on a and b to guarantee it works in the singular regime.

read the letter

This paper derives conditions for when the homogeneous equation ax - xb = 0 and the inhomogeneous ax - xb = c have solutions over the quaternions, then constructs the general and nonzero solutions by taking square roots of quaternions once those conditions hold. That explicit construction for the singular case is the main new piece. Prior work on quaternionic Sylvester equations has mostly stayed with the regular case or used different algebraic tools, so spelling out the square-root route adds a concrete method that people can try to apply directly. The derivations look straightforward and avoid obvious circularity or heavy fitting to prior results. The existence conditions are stated plainly enough from what is shown, and the focus on both homogeneous and inhomogeneous versions keeps the scope tight and useful. One soft spot is the square-root step itself. Quaternion square roots are non-unique for non-real elements, and the map does not automatically commute with left multiplication by a and right multiplication by b. In the singular regime the homogeneous operator already has a kernel, so a chosen square root has to satisfy extra commutation or conjugacy relations to recover all solutions or even to solve the inhomogeneous equation. The paper supplies existence conditions but does not appear to verify that every pair a, b meeting those conditions admits a square-root branch that preserves the required relations. If that verification is missing or only implicit, the formulas need tighter restrictions or a discussion of how to pick the root. This work is for readers who already work with quaternionic linear algebra, such as in specialized control or graphics settings where explicit solution formulas matter. A specialist who needs a tool for these particular equations will find the constructions worth checking. The paper shows clear algebraic engagement and is grounded enough to deserve referee time rather than a desk rejection.

Referee Report

2 major / 1 minor

Summary. The manuscript investigates the homogeneous quaternionic Sylvester equation ax − xb = 0 and the inhomogeneous equation ax − xb = c. It states conditions for the existence of solutions and derives explicit expressions for the general and nonzero solutions by applying quaternion square roots.

Significance. If the square-root constructions are valid without hidden restrictions, the explicit formulas would supply a direct, non-numerical route to solutions in the singular regime over the quaternions, extending classical Sylvester theory to a non-commutative division ring. The absence of fitted parameters or self-referential reductions is a methodological strength.

major comments (2)

[Derivation of solutions for the inhomogeneous equation] The derivation of general and nonzero solutions via quaternion square roots (following the existence conditions) does not verify that the chosen branch of √q commutes appropriately with left-multiplication by a and right-multiplication by b. Because square roots of non-real quaternions form a sphere and the map q ↦ √q is not a homomorphism, substitution into ax − xb = c can fail to recover solutions or satisfy the equation unless a and b obey an extra relation (e.g., Re(a) = Re(b) and |a| = |b| or conjugacy) not stated in the existence theorems.
[Derivation of solutions for the homogeneous equation] For the homogeneous case ax − xb = 0, the paper claims to obtain all solutions from square-root expressions once existence holds, yet the kernel of the operator is nontrivial precisely in the singular regime; it is unclear whether the square-root parametrization spans the full solution space or only a subset.

minor comments (1)

Notation for the quaternion square-root operation should be introduced with an explicit statement of the branch or selection criterion employed, especially when the argument is non-real.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript on solutions of singular Sylvester equations over the quaternions and for the constructive comments. We respond to each major comment below. Where clarifications are needed, we will revise the manuscript accordingly.

read point-by-point responses

Referee: [Derivation of solutions for the inhomogeneous equation] The derivation of general and nonzero solutions via quaternion square roots (following the existence conditions) does not verify that the chosen branch of √q commutes appropriately with left-multiplication by a and right-multiplication by b. Because square roots of non-real quaternions form a sphere and the map q ↦ √q is not a homomorphism, substitution into ax − xb = c can fail to recover solutions or satisfy the equation unless a and b obey an extra relation (e.g., Re(a) = Re(b) and |a| = |b| or conjugacy) not stated in the existence theorems.

Authors: The existence conditions in the paper are formulated so that a quaternion q arises for which a branch of its square root can be selected to satisfy the necessary commutation relations with a and b. Under these conditions the substitution recovers a solution to ax − xb = c. We will add an explicit verification paragraph after the construction, showing step-by-step that the chosen square root commutes appropriately with the left and right multiplications when the stated conditions hold, thereby confirming that no additional hidden relation on a and b is required beyond what is already assumed. revision: yes
Referee: [Derivation of solutions for the homogeneous equation] For the homogeneous case ax − xb = 0, the paper claims to obtain all solutions from square-root expressions once existence holds, yet the kernel of the operator is nontrivial precisely in the singular regime; it is unclear whether the square-root parametrization spans the full solution space or only a subset.

Authors: In the homogeneous case the square-root parametrization is constructed to be surjective onto the full solution set. The two-dimensional freedom in choosing the square root on the sphere precisely matches the dimension of the kernel when the existence condition is satisfied. We will insert a short completeness argument demonstrating that every solution x can be recovered from some choice of square root, confirming that the parametrization is exhaustive rather than a proper subset. revision: yes

Circularity Check

0 steps flagged

No circularity: direct algebraic derivation of existence conditions and explicit solutions via quaternion square roots

full rationale

The paper states existence conditions for the homogeneous equation ax−xb=0 and inhomogeneous ax−xb=c, then constructs general and nonzero solutions by applying quaternion square roots. This is a standard constructive approach in non-commutative algebra: the square-root step is introduced as an external algebraic operation once the existence criteria (presumably on the spectra or norms of a and b) are satisfied, without the solutions being used to define the square roots or vice versa. No self-citation load-bearing step, no fitted parameter renamed as prediction, and no ansatz smuggled from prior work by the same authors appears in the provided abstract or reader summary. The derivation chain therefore remains self-contained against external algebraic facts about quaternions.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard properties of the quaternion division ring and the existence of square roots within that ring for the singular case. No free parameters or invented entities are mentioned in the abstract.

axioms (1)

standard math Quaternions form a division ring in which square roots of certain elements can be defined and used to construct solutions to linear equations.
Invoked implicitly when the authors state that solutions are derived by applying quaternion square roots.

pith-pipeline@v0.9.0 · 5563 in / 1237 out tokens · 22130 ms · 2026-05-18T09:45:19.177930+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Conditions for the existence of solutions are provided. In addition, the general and nonzero solutions to these equations are derived applying quaternion square roots.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We present both the general and nonzero solutions of the homogeneous singular Sylvester equation in quaternions, establishing the connection to the quaternionic roots.

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