[Paper Review] A note on solution of $ax+xb=c$ by Clifford algebras

A. Dargys, A. Acus|arXiv (Cornell University)|Feb 25, 2019

Algebraic and Geometric Analysis6 references4 citations

TL;DR

This paper presents a coordinate-free method for solving the Sylvester equation $ax + xb = c$ in Clifford algebras $\mathrm{Cl}_{p,q}$ for $p+q \leq 3$ using grade negation and algebraic conjugation operations. It derives explicit solutions analogous to quaternionic methods by constructing scalar-like centers from multivectors, enabling inversion-based solutions when the central expressions are non-singular, with explicit computation demonstrated in $\mathrm{Cl}_{3,0}$.

ABSTRACT

The coordinate-free solutions of the multivector equation $ax+xb=c$ are discussed and presented for the Clifford algebras $Cl_{p,q}$ when $p+q\le 3$.

Motivation & Objective

• To develop a coordinate-free solution method for the Sylvester equation $ax + xb = c$ in low-dimensional Clifford algebras $\mathrm{Cl}_{p,q}$ with $p+q \leq 3$.
• To extend the quaternionic solution technique—based on scalar invariants from conjugation—to general Clifford algebras by identifying suitable algebraic conjugations that generate commuting centers.
• To determine which Clifford algebras allow such solutions by analyzing the existence of two independent central invariants: $a + \bar{a}$ and $a\bar{a}$.
• To demonstrate the method with explicit computation in $\mathrm{Cl}_{3,0}$, verifying the solution satisfies the original equation.

Proposed method

• Uses grade negation operation $a_{\bar{1},\bar{2}}$ to define a generalized conjugation that simultaneously inverts vectors and bivectors, replacing standard Clifford conjugation.
• Applies the same algebraic structure to $a$ and $b$, forming $a + \bar{a}$ and $a\bar{a}$ as elements of the center $\mathrm{Cen}(\mathrm{Cl}_{p,q})$ to ensure commutativity with all multivectors.
• Derives a solution formula analogous to the quaternionic case: $x = \left[(a^2 + b\bar{b}) + a(b + \bar{b})\right]^{-1}(ac + c\bar{b})$, valid when the inverse exists.
• Employs the inverse formula for 3D multivectors: $A^{-1} = \frac{C(AC)_{\bar{3}}}{AC(AC)_{\bar{3}}}$, where $C = A_{\bar{1},\bar{2}}$, to compute the inverse of the coefficient multivector.
• Verifies the solution by direct substitution into the original equation $ax + xb = c$ for a specific $\mathrm{Cl}_{3,0}$ example with given $a$, $b$, and $c$.
• Explores higher-dimensional algebras $\mathrm{Cl}_{n,0}$, $\mathrm{Cl}_{0,n}$, and $\mathrm{Cl}_{p,q}$ with $n=4,5,6$, but finds only one central invariant rather than two, limiting the method’s applicability.

Experimental results

Research questions

• RQ1 Which Clifford algebras $\mathrm{Cl}_{p,q}$ with $p+q \leq 3$ admit a coordinate-free solution to the Sylvester equation $ax + xb = c$ using a generalized conjugation?
• RQ2 Can the quaternionic solution technique—relying on scalar invariants $b + \bar{b}$ and $b\bar{b}$—be generalized to multivectors in $\mathrm{Cl}_{p,q}$ via grade negation?
• RQ3 What conditions must be satisfied for the solution formula $x = \left[(a^2 + b\bar{b}) + a(b + \bar{b})\right]^{-1}(ac + c\bar{b})$ to be valid in $\mathrm{Cl}_{p,q}$?
• RQ4 How does the structure of the center $\mathrm{Cen}(\mathrm{Cl}_{p,q})$, particularly the presence of scalar and pseudoscalar components, affect the solvability of the equation?
• RQ5 Why does the method fail to extend to $\mathrm{Cl}_{1,3}$ and $\mathrm{Cl}_{3,1}$ despite their isomorphism to even subalgebras of $\mathrm{Cl}_{4,0}$?

Key findings

• The method successfully solves $ax + xb = c$ in $\mathrm{Cl}_{2,0}$, $\mathrm{Cl}_{1,1}$, $\mathrm{Cl}_{0,2}$, $\mathrm{Cl}_{3,0}$, $\mathrm{Cl}_{2,1}$, $\mathrm{Cl}_{1,2}$, and $\mathrm{Cl}_{0,3}$ using the generalized conjugation $a_{\bar{1},\bar{2}}$ and the derived solution formula.
• The solution is singular when the coefficient multivector $(a^2 + b\bar{b}) + a(b + \bar{b})$ is zero, which occurs when $a + \bar{a} = 0$ in the case $b = a$, indicating a loss of invertibility.
• For $\mathrm{Cl}_{3,0}$, the method yields a concrete solution: $x = \frac{1}{2177719}(359677 + 601305\mathbf{e}_1 - 155957\mathbf{e}_2 - 436078\mathbf{e}_3 + 209677\mathbf{e}_{12} + 1076362\mathbf{e}_{13} - 489350\mathbf{e}_{23} + 27015\mathbf{e}_{123})$, which satisfies the original equation.
• The inverse of the coefficient multivector is computed using the formula $A^{-1} = \frac{C(AC)_{\bar{3}}}{AC(AC)_{\bar{3}}}$ with $C = A_{\bar{1},\bar{2}}$, and the result is verified numerically.
• In higher-dimensional algebras ($n=4,5,6$), only one central invariant ($a + \bar{a}$ or $a\bar{a}$) is found, but not both, preventing the method’s generalization beyond $p+q \leq 3$.
• The method is applicable to $\mathrm{Cl}_{1,3}$ and $\mathrm{Cl}_{3,1}$ only via their even subalgebras, but fails due to insufficient central structure, limiting its use in relativistic physics applications.