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Basis-free solution to Sylvester equation in Clifford algebra of arbitrary dimension. (English) Zbl 1476.15029

Summary: 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.

MSC:

15A24 Matrix equations and identities
15A66 Clifford algebras, spinors
15A67 Applications of Clifford algebras to physics, etc.
15A09 Theory of matrix inversion and generalized inverses
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