Department of Mathematics
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Ore matrices are matrices over Ore algebras (including differential operators and difference operators), which have a long research history, at least dated back to Jacobson's seminal work in 1940s. In past two decades, Ore matrices have attracted more and more people in computer algebra area, and many important properties of Ore matrices have been discussed by using symbolic computation methods.

It is well-known that the generalized inverse of matrices over commutative rings (or fields), especially the Moore-Penrose inverse, play important roles in matrix theory and have applications in many areas: solving matrix equations, statistics, engineering, etc. This motivates us to consider the generalized inverse of Ore matrices.

First we define the generalized inverse and the Moore-Penrose inverse for Ore matrices, and prove some basic properties including uniqueness. Unlike the commutative case, the generalized inverse for a given Ore matrix may not exist. We use blocked matrices and greatest common right (left) divisor computations to give some sufficient and necessary conditions for Ore matrices to have the generalized inverses and the Moore-Penrose inverses. Moreover when these inverses exist, we construct algorithms to compute them.