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About the uniqueness solution of the matrix polynomial equation $A(\lambda)X(\lambda)-Y(\lambda)B(\lambda)=C(\lambda)$. (English) Zbl 1176.15019
Sufficient conditions are provided for the uniqueness of the solution of the matrix polynomial equation $$A(\lambda)X(\lambda)-Y(\lambda)B(\lambda)=C(\lambda)\tag1$$ over an arbitrary field $F$, where $A(\lambda)\in F^{m\times m}[\lambda]$, $B(\lambda)\in F^{n\times n}[\lambda]$, $C(\lambda)\in F^{m\times n}[\lambda]$. It is assumed that the matrix $B(\lambda)$ is left equivalent to a regular polynomial matrix (i.e. $B(\lambda)$ admits the representation $B(\lambda)=W(\lambda)D(\lambda)$ where $W(\lambda)\in GL(n,F(\lambda)$ and $D(\lambda)\in F^{n\times n}[\lambda]$ is a regular polynomial matrix with $\deg D(\lambda)<\deg B(\lambda)$). The main result states that if $(\det A(\lambda),\det A(\lambda))=1$, then the matrix equation (1) has a unique solution $\{X_0(\lambda),Y_0(\lambda)\}$ such that $\deg X_0(\lambda)<\deg D(\lambda)$. A similar result is obtained for $A(\lambda)$ right equivalent to a regular polynomial matrix.

15A24Matrix equations and identities
15A54Matrices over function rings
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