Basis-free Solution to Sylvester Equation in Clifford Algebra of Arbitrary Dimension

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