Define / method for adjoint LU rhs. (#33209)

Fixes #33177

(cherry picked from commit 4c8f8fa)

stdlib/LinearAlgebra/src/lu.jl

@@ -416,13 +416,29 @@ function ldiv!(adjA::Adjoint{<:Any,<:LU{<:Any,<:StridedMatrix}}, B::StridedVecOr
     _apply_inverse_ipiv_rows!(A, B)
 end
 
-\(A::Adjoint{<:Any,<:LU}, B::Adjoint{<:Any,<:StridedVecOrMat}) = A \ copy(B)
-\(A::Transpose{<:Any,<:LU}, B::Transpose{<:Any,<:StridedVecOrMat}) = A \ copy(B)
-\(A::Adjoint{T,<:LU{T,<:StridedMatrix}}, B::Adjoint{T,<:StridedVecOrMat{T}}) where {T<:BlasComplex} =
+(\)(A::Adjoint{<:Any,<:LU}, B::Adjoint{<:Any,<:StridedVecOrMat}) = A \ copy(B)
+(\)(A::Transpose{<:Any,<:LU}, B::Transpose{<:Any,<:StridedVecOrMat}) = A \ copy(B)
+(\)(A::Adjoint{T,<:LU{T,<:StridedMatrix}}, B::Adjoint{T,<:StridedVecOrMat{T}}) where {T<:BlasComplex} =
     LAPACK.getrs!('C', A.parent.factors, A.parent.ipiv, copy(B))
-\(A::Transpose{T,<:LU{T,<:StridedMatrix}}, B::Transpose{T,<:StridedVecOrMat{T}}) where {T<:BlasFloat} =
+(\)(A::Transpose{T,<:LU{T,<:StridedMatrix}}, B::Transpose{T,<:StridedVecOrMat{T}}) where {T<:BlasFloat} =
     LAPACK.getrs!('T', A.parent.factors, A.parent.ipiv, copy(B))
 
+function (/)(A::AbstractMatrix, F::Adjoint{<:Any,<:LU})
+    T = promote_type(eltype(A), eltype(F))
+    return adjoint(ldiv!(F.parent, copy_oftype(adjoint(A), T)))
+end
+# To avoid ambiguities with definitions in adjtrans.jl and factorizations.jl
+(/)(adjA::Adjoint{<:Any,<:AbstractVector}, F::Adjoint{<:Any,<:LU}) = adjoint(F.parent \ adjA.parent)
+(/)(adjA::Adjoint{<:Any,<:AbstractMatrix}, F::Adjoint{<:Any,<:LU}) = adjoint(F.parent \ adjA.parent)
+function (/)(trA::Transpose{<:Any,<:AbstractVector}, F::Adjoint{<:Any,<:LU})
+    T = promote_type(eltype(trA), eltype(F))
+    return adjoint(ldiv!(F.parent, convert(AbstractVector{T}, conj(trA.parent))))
+end
+function (/)(trA::Transpose{<:Any,<:AbstractMatrix}, F::Adjoint{<:Any,<:LU})
+    T = promote_type(eltype(trA), eltype(F))
+    return adjoint(ldiv!(F.parent, convert(AbstractMatrix{T}, conj(trA.parent))))
+end
+
 function det(F::LU{T}) where T
     n = checksquare(F)
     issuccess(F) || return zero(T)

stdlib/LinearAlgebra/test/lu.jl

@@ -316,4 +316,20 @@ end
                0 0 0 0 1 1 0 1]) ≈ 6
 end
 
+@testset "Issue #33177. No ldiv!(LU, Adjoint)" begin
+    A = [1 0; 1 1]
+    B = [1 2; 2 8]
+    F = lu(B)
+    @test (A  / F') * B == A
+    @test (A' / F') * B == A'
+
+    a = complex.(randn(2), randn(2))
+    @test (a' / F') * B ≈ a'
+    @test (transpose(a) / F') * B ≈ transpose(a)
+
+    A = complex.(randn(2, 2), randn(2, 2))
+    @test (A' / F') * B ≈ A'
+    @test (transpose(A) / F') * B ≈ transpose(A)
+end
+
 end # module TestLU