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Add is_nilpotent(::MatrixElem) #1783

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merged 4 commits into from
Sep 11, 2024
Merged

Add is_nilpotent(::MatrixElem) #1783

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e0bd091
Add `is_nilpotent(::MatrixElem)`
lgoettgens Sep 6, 2024
28f6068
Bump version to 0.42.5
lgoettgens Sep 6, 2024
553bb9b
Add tr shortcut
lgoettgens Sep 6, 2024
1bbf5b6
Update src/Matrix.jl
lgoettgens Sep 10, 2024
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2 changes: 1 addition & 1 deletion Project.toml
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Original file line number Diff line number Diff line change
@@ -1,6 +1,6 @@
name = "AbstractAlgebra"
uuid = "c3fe647b-3220-5bb0-a1ea-a7954cac585d"
version = "0.42.4"
version = "0.42.5"

[deps]
InteractiveUtils = "b77e0a4c-d291-57a0-90e8-8db25a27a240"
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6 changes: 6 additions & 0 deletions docs/src/matrix.md
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Expand Up @@ -916,6 +916,12 @@ det{T <: RingElem}(::MatElem{T})
rank{T <: RingElem}(::MatElem{T})
```

### Nilpotency

```@docs
is_nilpotent(::MatrixElem{T}) where {T <: RingElement}
```

### Minors

```@docs
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27 changes: 27 additions & 0 deletions src/Matrix.jl
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Expand Up @@ -3686,6 +3686,33 @@ function nullspace(M::MatElem{T}) where {T <: FieldElement}
return nullity, X
end

###############################################################################
#
# Nilpotency
#
###############################################################################

@doc raw"""
is_nilpotent(A::MatrixElem{T}) where {T <: RingElement}

Return if `A` is nilpotent, i.e. if there exists a natural number $k$
such that $A^k = 0$. If `A` is not square an exception is raised.
"""
function is_nilpotent(A::MatrixElem{T}) where {T <: RingElement}
!is_square(A) && error("Dimensions don't match in is_nilpotent")
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is_zero(tr(A)) || return false
n = nrows(A)
A = deepcopy(A)
i = 1
is_zero(A) && return true
while i < n
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Actually, is this algorithm even correct when A contains zero divisors? I normally think about matrices over integral domains, and then of course A is nilpotent iff A^n=0. But over a ring with zero-divisors, we could have a triangularizable matrix with nilpotent eigenvalues where $A^n\neq0$ but still $A^k=0$ for some $k&gt;n$.

Admittedly this is somewhat fringe, but we should still handle it right? For starters we could add is_domain_type(T) || error("Only supported over integral domains") ? (AFAIK there is no is_domain / is_integral_domain which for e.g. $\mathbb{Z}/n\mathbb{Z}$ takes the modulus $n$ into account)

Then if someone needs this over non-domains, they at least get a helpful error and can provide an implementation for their use case.

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Good catch! Thanks for mentioning this. I added the suggested error

i *= 2
A = mul!(A, A, A)
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It might or might not be faster to compute the minimal or characteristic polynomial (if there are dedicated optimized methods for this and the given matrix type

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yeah that's true. but for that one would need to know which specialized methods exist in which cases, which I honestly don't

is_zero(A) && return true
end
return false
end

###############################################################################
#
# Hessenberg form
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1 change: 1 addition & 0 deletions src/exports.jl
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Expand Up @@ -294,6 +294,7 @@ export is_monic
export is_monomial
export is_monomial_recursive
export is_negative
export is_nilpotent
export is_odd
export is_one
export is_perfect
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7 changes: 7 additions & 0 deletions test/Matrix-test.jl
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Expand Up @@ -64,6 +64,13 @@ end
@test is_upper_triangular(matrix(ZZ, A))
end

@testset "Matrix.is_nilpotent" begin
@test is_nilpotent(zero_matrix(QQ, 3, 3))
@test is_nilpotent(upper_triangular_matrix(QQ.([0,1,1,0,1,0])))
@test !is_nilpotent(identity_matrix(QQ, 3))
@test !is_nilpotent(diagonal_matrix(QQ, [1,-1,0]))
end

@testset "Matrix.concat" begin
for R in [ZZ, QQ, polynomial_ring(QQ, [:x, :y])[1]]
S = matrix_space(R, 3, 3)
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