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]
| n\m | 1 | 2 | 3 | 4 | 5 | 6 |
|---|---|---|---|---|---|---|
| 2 | ✅ | ✅ | ||||
| 3 | ✅ | ✖️ | ||||
| 4 | ✅ | ✖️ | ||||
| 5 | ✅ | ✖️ | ✖️ | ✖️ | ||
| ... | ✅ | ✖️ | ✖️ | ✖️ | ||
| 9 | ✅ | ✖️ | ✖️ | ✖️ | ✖️ | |
| ... | ✅ | ✖️ | ✖️ | ✖️ | ✖️ | |
| 15 | ✅ | ✖️ | ✖️ | ✖️ | ✖️ | ✖️ |
References
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- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- ^ Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
- Lua error in Module:Citation/CS1/Configuration at line 2172: attempt to index field '?' (a nil value).
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