Basis-free solution to Sylvester equation in Clifford algebra of arbitrary dimension

TL;DR

This paper introduces a basis-free method for solving the Sylvester equation within Clifford algebra of any dimension, utilizing geometric algebra operations and extending formulas for specific cases.

Contribution

It provides the first basis-free solutions to the Sylvester equation in Clifford algebra for arbitrary dimensions, including new formulas for n=4 and proofs for n=5.

Findings

Basis-free solutions involve only Clifford product, summation, and conjugation.

New formulas for n=4 case and proofs for n=5 case.

Applicable to symbolic computation in various fields.

Abstract

The Sylvester equation and its particular case, the Lyapunov equation, are widely used in image processing, control theory, stability analysis, signal processing, model reduction, and many more. We present basis-free solution to the Sylvester equation in Clifford (geometric) algebra of arbitrary dimension. The basis-free solutions involve only the operations of Clifford (geometric) product, summation, and the operations of conjugation. To obtain the results, we use the concepts of characteristic polynomial, determinant, adjugate, and inverse in Clifford algebras. For the first time, we give alternative formulas for the basis-free solution to the Sylvester equation in the case $n=4$, the proofs for the case $n=5$ and the case of arbitrary dimension $n$. The results can be used in symbolic computation.