diff --git a/stdlib/LinearAlgebra/src/diagonal.jl b/stdlib/LinearAlgebra/src/diagonal.jl
index d849640a351f1..e4d4615c1a2c5 100644
--- a/stdlib/LinearAlgebra/src/diagonal.jl
+++ b/stdlib/LinearAlgebra/src/diagonal.jl
@@ -316,7 +316,7 @@ function mul!(C::AbstractMatrix, Da::Diagonal, Db::Diagonal, alpha::Number, beta
     return C
 end
 
-#TODO: many of /, \ related function has no size check and singular check
+#TODO: many of /, \ related function has no singular check
 (/)(A::AbstractVecOrMat, D::Diagonal) =
     rdiv!((typeof(oneunit(eltype(D))/oneunit(eltype(A)))).(A), D)
 (/)(Da::Diagonal, Db::Diagonal) = Diagonal(Da.diag ./ Db.diag)
@@ -345,6 +345,7 @@ end
 (\)(D::Diagonal, b::AbstractVector) = D.diag .\ b
 (\)(Da::Diagonal, Db::Diagonal) = Diagonal(Da.diag .\ Db.diag)
 
+#TODO: we should check size(x,2) == size(b,2)
 ldiv!(x::AbstractVecOrMat, A::Diagonal, b::AbstractVecOrMat) = (x .= A.diag .\ b)
 
 function ldiv!(D::Diagonal, B::AbstractVecOrMat)
@@ -548,11 +549,17 @@ function svd(D::Diagonal{<:Number})
     return SVD(Up, S[piv], copy(Vp'))
 end
 
-# disambiguation methods: * of Diagonal and Adj/Trans AbsVec
-*(x::Adjoint{<:Any,<:AbstractVector}, D::Diagonal) = Adjoint(map((t,s) -> t'*s, D.diag, parent(x)))
-*(x::Transpose{<:Any,<:AbstractVector}, D::Diagonal) = Transpose(map((t,s) -> transpose(t)*s, D.diag, parent(x)))
-*(x::Adjoint{<:Any,<:AbstractVector},   D::Diagonal, y::AbstractVector) = _mapreduce_prod(*, x, D, y)
-*(x::Transpose{<:Any,<:AbstractVector}, D::Diagonal, y::AbstractVector) = _mapreduce_prod(*, x, D, y)
+# disambiguation methods: * and / of Diagonal and Adj/Trans AbsVec
+*(x::AdjointAbsVec, D::Diagonal) = Adjoint(map((t,s) -> t'*s, D.diag, parent(x)))
+*(x::TransposeAbsVec, D::Diagonal) = Transpose(map((t,s) -> transpose(t)*s, D.diag, parent(x)))
+*(x::AdjointAbsVec,   D::Diagonal, y::AbstractVector) = _mapreduce_prod(*, x, D, y)
+*(x::TransposeAbsVec, D::Diagonal, y::AbstractVector) = _mapreduce_prod(*, x, D, y)
+/(u::AdjointAbsVec, D::Diagonal) = adjoint(adjoint(D) \ u.parent)
+/(u::TransposeAbsVec, D::Diagonal) = transpose(transpose(D) \ u.parent)
+# disambiguation methods: Call unoptimized version for user defined AbstractTriangular.
+*(A::AbstractTriangular, D::Diagonal) = Base.@invoke *(A::AbstractMatrix, D::Diagonal)
+*(D::Diagonal, A::AbstractTriangular) = Base.@invoke *(D::Diagonal, A::AbstractMatrix)
+
 dot(x::AbstractVector, D::Diagonal, y::AbstractVector) = _mapreduce_prod(dot, x, D, y)
 
 dot(A::Diagonal, B::Diagonal) = dot(A.diag, B.diag)
@@ -620,4 +627,4 @@ end
 
 function Base.muladd(A::Diagonal, B::Diagonal, z::Diagonal)
     Diagonal(A.diag .* B.diag .+ z.diag)
-end
+end
\ No newline at end of file