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. © Springer-Verlag Berlin Heidelberg 2005.
CITATION STYLE
Bell, P., & Potapov, I. (2005). On the membership of invertible diagonal matrices. In Lecture Notes in Computer Science (Vol. 3572, pp. 146–157). Springer Verlag. https://doi.org/10.1007/11505877_13
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