Abstract
In this paper we consider decidability questions that are related to the membership problem in matrix semigroups. In particular we consider the membership of a particular invertible diagonal matrix in a matrix semigroup and then a scalar matrix, which has a separate geometric interpretation. Both problems have been open for any dimensions and are shown to be undecidable in dimenesion 4 with integral matrices and in dimension 3 with rational matrices by a reduction of the Post Correspondence Problem (PCP). Although the idea of PCP reduction is standard for such problems, we suggest a new coding technique to cover the case of diagonal matrices.
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Bell, P., Potapov, I. (2005). On the Membership of Invertible Diagonal Matrices. In: De Felice, C., Restivo, A. (eds) Developments in Language Theory. DLT 2005. Lecture Notes in Computer Science, vol 3572. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11505877_13
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DOI: https://doi.org/10.1007/11505877_13
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26546-7
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