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On solutions of generalized Sylvester equation in polynomial matrices. (English) Zbl 1398.65078

Summary: This paper deals with the generalized Sylvester equation in polynomial matrices \(A(\lambda)X(\lambda)+Y(\lambda)B(\lambda)=C(\lambda)\), where \(A(\lambda)\) and \(B(\lambda)\) are monic. If the equation has solutions, then it has a solution satisfying a natural degree constraint condition. It is shown that the generalized Sylvester equation in polynomial matrices can be reduced to the linear matrix equation \(A_R^nY+\sum_{i=0}^{n-1}A_R^iYB_i=C\), where \(A_R\) is the second block-companion matrix of \(A(\lambda)\).

MSC:

65F30 Other matrix algorithms (MSC2010)
15A24 Matrix equations and identities
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