Matrix mortality problem

In computer science, the matrix mortality problem (or mortal matrix problem) is a decision problem that asks, given a set of size m of n×n matrices with integer coefficients, whether the zero matrix can be expressed as a finite product of matrices from this set.

The matrix mortality problem is known to be undecidable when n ≥ 3.[1] In fact, it is already undecidable for sets of 6 matrices (or more) when n = 3, for 4 matrices when n = 5, for 3 matrices when n = 9, and for 2 matrices when n = 15.[2]

In the case n = 2, it is an open problem whether matrix mortality is decidable, but several special cases have been solved: the problem is decidable for sets of 2 matrices,[3] and for sets of matrices which contain at most one invertible matrix.[4]

The current frontier of knowledge
n\m 1 2 3 4 5 6
2 ✅ ✅
3 ✅ ✖️
4 ✅ ✖️
5 ✅ ✖️ ✖️ ✖️
... ✅ ✖️ ✖️ ✖️
9 ✅ ✖️ ✖️ ✖️ ✖️
... ✅ ✖️ ✖️ ✖️ ✖️
15 ✅ ✖️ ✖️ ✖️ ✖️ ✖️

References

  1. ^ Paterson, Michael S. (1970). "Unsolvability in 3 × 3 matrices". Studies in Applied Mathematics. 49: 105–107. doi:10.1002/sapm1970491105. MR 0255400.
  2. ^ Cassaigne, Julien; Halava, Vesa; Harju, Tero; Nicolas, Francois (2014). "Tighter Undecidability Bounds for Matrix Mortality, Zero-in-the-Corner Problems, and More". arXiv:1404.0644 [cs.DM].
  3. ^ Bournez, Olivier; Branicky, Michael (2002). "The Mortality Problem for Matrices of Low Dimensions" (PDF). Theory of Computing Systems. 35 (4): 433–448. doi:10.1007/s00224-002-1010-5.
  4. ^ Heckman, Christopher Carl (2019). "The 2×2 Matrix Mortality Problem and Invertible Matrices". arXiv:1912.09991 [math.RA].
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