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.

Sylvester 方程及其特殊情况 Lyapunov 方程广泛用于图像处理、控制理论、稳定性分析、信号处理、模型简化等。我们提出了任意维数的 Clifford（几何）代数中 Sylvester 方程的无基解。无基解仅涉及 Clifford（几何）乘积运算、求和运算和共轭运算。为了得到结果，我们在 Clifford 代数中使用了特征多项式、行列式、调节和逆的概念。我们第一次给出了 Sylvester 方程在\(n=4\)情况下的无基解的替代公式、\(n=5\)情况和任意维数n情况的证明. 结果可用于符号计算。