The Mortality Problem for Matrices of Low Dimensions

Abstract

In this paper we discuss the existence of an algorithm to decide if a given set of 2 \times 2 matrices is mortal. A set F={A ₁ ,\ldots,A _m } of square matrices is said to be mortal if there exist an integer k \geq 1 and some integers i ₁ ,i ₂ ,\ldots,i _k ∈ {1, \ldots, m} with A _i1 A _i2 ⋅s A _ik =0 . We survey this problem and propose some new extensions. We prove the problem to be BSS-undecidable for real matrices and Turing-decidable for two rational matrices. We relate the problem for rational matrices to the entry-equivalence problem, to the zero-in-the-corner problem, and to the reachability problem for piecewise-affine functions. Finally, we state some NP-completeness results.