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Adjoint/transpose for tridiagonal matrices preserve structure #1370

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Adjoint/transpose for tridiagonal matrices preserve structure #1370

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15 changes: 3 additions & 12 deletions src/tridiag.jl
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Original file line number Diff line number Diff line change
Expand Up @@ -174,16 +174,13 @@ end
isreal(S::SymTridiagonal) = isreal(S.dv) && isreal(S.ev)

transpose(S::SymTridiagonal) = S
adjoint(S::SymTridiagonal{<:Number}) = SymTridiagonal(vec(adjoint(S.dv)), vec(adjoint(S.ev)))
adjoint(S::SymTridiagonal{<:Number, <:Base.ReshapedArray{<:Number,1,<:Adjoint}}) =
SymTridiagonal(adjoint(parent(S.dv)), adjoint(parent(S.ev)))
adjoint(S::SymTridiagonal) = SymTridiagonal(_vecadjoint(S.dv), _vecadjoint(S.ev))
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@stevengj stevengj Jun 2, 2025 •
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I think this is only correct for SymTridiagonal{<:Number}.

In particular, the off-diagonal elements should be adjoint.(transpose.(S.ev)) or equivalently conj(S.ev). The reason is that S.ev stores the upper diagonal, but when you take the adjoint, the new upper diagonal comes from the adjoint of the lower diagonal, but the lower diagonal of S is transpose.(S.ev).

(Why we allow SymTridiagonal for non-Number elements is beyond me.)


permutedims(S::SymTridiagonal) = S
function permutedims(S::SymTridiagonal, perm)
Base.checkdims_perm(axes(S), axes(S), perm)
NTuple{2}(perm) == (2, 1) ? permutedims(S) : S
end
Base.copy(S::Adjoint{<:Any,<:SymTridiagonal}) = SymTridiagonal(map(x -> copy.(adjoint.(x)), (S.parent.dv, S.parent.ev))...)

ishermitian(S::SymTridiagonal) = isreal(S.dv) && isreal(_evview(S))
issymmetric(S::SymTridiagonal) = true
Expand Down Expand Up @@ -683,23 +680,17 @@ for func in (:conj, :copy, :real, :imag)
end
isreal(T::Tridiagonal) = isreal(T.dl) && isreal(T.d) && isreal(T.du)

adjoint(S::Tridiagonal{<:Number}) = Tridiagonal(vec(adjoint(S.du)), vec(adjoint(S.d)), vec(adjoint(S.dl)))
adjoint(S::Tridiagonal{<:Number, <:Base.ReshapedArray{<:Number,1,<:Adjoint}}) =
Tridiagonal(adjoint(parent(S.du)), adjoint(parent(S.d)), adjoint(parent(S.dl)))
transpose(S::Tridiagonal{<:Number}) = Tridiagonal(S.du, S.d, S.dl)
adjoint(S::Tridiagonal) = Tridiagonal(_vecadjoint(S.du), _vecadjoint(S.d), _vecadjoint(S.dl))
transpose(S::Tridiagonal) = Tridiagonal(_vectranspose(S.du), _vectranspose(S.d), _vectranspose(S.dl))
permutedims(T::Tridiagonal) = Tridiagonal(T.du, T.d, T.dl)
function permutedims(T::Tridiagonal, perm)
Base.checkdims_perm(axes(T), axes(T), perm)
NTuple{2}(perm) == (2, 1) ? permutedims(T) : T
end
Base.copy(aS::Adjoint{<:Any,<:Tridiagonal}) = (S = aS.parent; Tridiagonal(map(x -> copy.(adjoint.(x)), (S.du, S.d, S.dl))...))
Base.copy(tS::Transpose{<:Any,<:Tridiagonal}) = (S = tS.parent; Tridiagonal(map(x -> copy.(transpose.(x)), (S.du, S.d, S.dl))...))

ishermitian(S::Tridiagonal) = all(ishermitian, S.d) && all(Iterators.map((x, y) -> x == y', S.du, S.dl))
issymmetric(S::Tridiagonal) = all(issymmetric, S.d) && all(Iterators.map((x, y) -> x == transpose(y), S.du, S.dl))

\(A::Adjoint{<:Any,<:Tridiagonal}, B::Adjoint{<:Any,<:AbstractVecOrMat}) = copy(A) \ B

function diag(M::Tridiagonal, n::Integer=0)
# every branch call similar(..., ::Int) to make sure the
# same vector type is returned independent of n
Expand Down
22 changes: 22 additions & 0 deletions test/tridiag.jl
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Expand Up @@ -1207,4 +1207,26 @@ end
@test_throws BoundsError S[LinearAlgebra.BandIndex(0,size(S,1)+1)]
end

@testset "lazy adjtrans" begin
dv = fill([1 2; 3 4], 3)
ev = fill([5 6; 7 8], 2)
T = Tridiagonal(ev, dv, ev)
S = SymTridiagonal(dv, ev)
m = [2 4; 4 2]
for B in (copy(T), copy(S))
for op in (transpose, adjoint)
C = op(B)
el = op(m)
C[1,1] = el
@test B[1,1] == m
if B isa Tridiagonal
C[2,1] = el
@test B[1,2] == m
end
@test (@allocated op(B)) == 0
@test (@allocated op(op(B))) == 0
end
end
end

end # module TestTridiagonal