Mortality in Matrix Semigroups

Abstract

We present a new shorter and simplified proof for the undecidability of the mor- tality problem in matrix semigroups, originally proved by Paterson in 1970. We use the clever coding technique introduced by Paterson to achieve also a new res- ult, the undecidability of the vanishing (left) upper corner. Since our proof for the undecidability of the mortality problem uses only 8 matrices, a new bound for the dimension for the undecidability of the mortality in the two generator matrix semigroup is achieved.