diff --git a/stdlib/LinearAlgebra/src/LinearAlgebra.jl b/stdlib/LinearAlgebra/src/LinearAlgebra.jl
index 10cc9a2f3459a..a80c8d0600a78 100644
--- a/stdlib/LinearAlgebra/src/LinearAlgebra.jl
+++ b/stdlib/LinearAlgebra/src/LinearAlgebra.jl
@@ -523,6 +523,17 @@ _makevector(x::AbstractVector) = Vector(x)
 _pushzero(A) = (B = similar(A, length(A)+1); @inbounds B[begin:end-1] .= A; @inbounds B[end] = zero(eltype(B)); B)
 _droplast!(A) = deleteat!(A, lastindex(A))
 
+# destination type for matmul
+matprod_dest(A::StructuredMatrix, B::StructuredMatrix, TS) = similar(B, TS, size(B))
+matprod_dest(A, B::StructuredMatrix, TS) = similar(A, TS, size(A))
+matprod_dest(A::StructuredMatrix, B, TS) = similar(B, TS, size(B))
+matprod_dest(A::StructuredMatrix, B::Diagonal, TS) = similar(A, TS)
+matprod_dest(A::Diagonal, B::StructuredMatrix, TS) = similar(B, TS)
+matprod_dest(A::Diagonal, B::Diagonal, TS) = similar(B, TS)
+matprod_dest(A::HermOrSym, B::Diagonal, TS) = similar(A, TS, size(A))
+matprod_dest(A::Diagonal, B::HermOrSym, TS) = similar(B, TS, size(B))
+
+# TODO: remove once not used anymore in SparseArrays.jl
 # some trait like this would be cool
 # onedefined(::Type{T}) where {T} = hasmethod(one, (T,))
 # but we are actually asking for oneunit(T), that is, however, defined for generic T as
diff --git a/stdlib/LinearAlgebra/src/bidiag.jl b/stdlib/LinearAlgebra/src/bidiag.jl
index 88cc145835df9..2b4b2ac98317c 100644
--- a/stdlib/LinearAlgebra/src/bidiag.jl
+++ b/stdlib/LinearAlgebra/src/bidiag.jl
@@ -802,34 +802,35 @@ ldiv!(c::AbstractVecOrMat, A::AdjOrTrans{<:Any,<:Bidiagonal}, b::AbstractVecOrMa
     (t = wrapperop(A); _rdiv!(t(c), t(b), t(A)); return c)
 
 ### Generic promotion methods and fallbacks
-\(A::Bidiagonal, B::AbstractVecOrMat) = ldiv!(_initarray(\, eltype(A), eltype(B), B), A, B)
+\(A::Bidiagonal, B::AbstractVecOrMat) =
+    ldiv!(matprod_dest(A, B, promote_op(\, eltype(A), eltype(B))), A, B)
 \(xA::AdjOrTrans{<:Any,<:Bidiagonal}, B::AbstractVecOrMat) = copy(xA) \ B
 
 ### Triangular specializations
 for tri in (:UpperTriangular, :UnitUpperTriangular)
     @eval function \(B::Bidiagonal, U::$tri)
-        A = ldiv!(_initarray(\, eltype(B), eltype(U), U), B, U)
+        A = ldiv!(matprod_dest(B, U, promote_op(\, eltype(B), eltype(U))), B, U)
         return B.uplo == 'U' ? UpperTriangular(A) : A
     end
     @eval function \(U::$tri, B::Bidiagonal)
-        A = ldiv!(_initarray(\, eltype(U), eltype(B), U), U, B)
+        A = ldiv!(matprod_dest(U, B, promote_op(\, eltype(U), eltype(B))), U, B)
         return B.uplo == 'U' ? UpperTriangular(A) : A
     end
 end
 for tri in (:LowerTriangular, :UnitLowerTriangular)
     @eval function \(B::Bidiagonal, L::$tri)
-        A = ldiv!(_initarray(\, eltype(B), eltype(L), L), B, L)
+        A = ldiv!(matprod_dest(B, L, promote_op(\, eltype(B), eltype(L))), B, L)
         return B.uplo == 'L' ? LowerTriangular(A) : A
     end
     @eval function \(L::$tri, B::Bidiagonal)
-        A = ldiv!(_initarray(\, eltype(L), eltype(B), L), L, B)
+        A = ldiv!(matprod_dest(L, B, promote_op(\, eltype(L), eltype(B))), L, B)
         return B.uplo == 'L' ? LowerTriangular(A) : A
     end
 end
 
 ### Diagonal specialization
 function \(B::Bidiagonal, D::Diagonal)
-    A = ldiv!(_initarray(\, eltype(B), eltype(D), D), B, D)
+    A = ldiv!(similar(D, promote_op(\, eltype(B), eltype(D)), size(D)), B, D)
     return B.uplo == 'U' ? UpperTriangular(A) : LowerTriangular(A)
 end
 
@@ -879,33 +880,34 @@ rdiv!(A::AbstractMatrix, B::AdjOrTrans{<:Any,<:Bidiagonal}) = @inline _rdiv!(A,
 _rdiv!(C::AbstractMatrix, A::AbstractMatrix, B::AdjOrTrans{<:Any,<:Bidiagonal}) =
     (t = wrapperop(B); ldiv!(t(C), t(B), t(A)); return C)
 
-/(A::AbstractMatrix, B::Bidiagonal) = _rdiv!(_initarray(/, eltype(A), eltype(B), A), A, B)
+/(A::AbstractMatrix, B::Bidiagonal) =
+    _rdiv!(similar(A, promote_op(/, eltype(A), eltype(B)), size(A)), A, B)
 
 ### Triangular specializations
 for tri in (:UpperTriangular, :UnitUpperTriangular)
     @eval function /(U::$tri, B::Bidiagonal)
-        A = _rdiv!(_initarray(/, eltype(U), eltype(B), U), U, B)
+        A = _rdiv!(matprod_dest(U, B, promote_op(/, eltype(U), eltype(B))), U, B)
         return B.uplo == 'U' ? UpperTriangular(A) : A
     end
     @eval function /(B::Bidiagonal, U::$tri)
-        A = _rdiv!(_initarray(/, eltype(B), eltype(U), U), B, U)
+        A = _rdiv!(matprod_dest(B, U, promote_op(/, eltype(B), eltype(U))), B, U)
         return B.uplo == 'U' ? UpperTriangular(A) : A
     end
 end
 for tri in (:LowerTriangular, :UnitLowerTriangular)
     @eval function /(L::$tri, B::Bidiagonal)
-        A = _rdiv!(_initarray(/, eltype(L), eltype(B), L), L, B)
+        A = _rdiv!(matprod_dest(L, B, promote_op(/, eltype(L), eltype(B))), L, B)
         return B.uplo == 'L' ? LowerTriangular(A) : A
     end
     @eval function /(B::Bidiagonal, L::$tri)
-        A = _rdiv!(_initarray(/, eltype(B), eltype(L), L), B, L)
+        A = _rdiv!(matprod_dest(B, L, promote_op(/, eltype(B), eltype(L))), B, L)
         return B.uplo == 'L' ? LowerTriangular(A) : A
     end
 end
 
 ### Diagonal specialization
 function /(D::Diagonal, B::Bidiagonal)
-    A = _rdiv!(_initarray(/, eltype(D), eltype(B), D), D, B)
+    A = _rdiv!(similar(D, promote_op(/, eltype(D), eltype(B)), size(D)), D, B)
     return B.uplo == 'U' ? UpperTriangular(A) : LowerTriangular(A)
 end
 
diff --git a/stdlib/LinearAlgebra/src/diagonal.jl b/stdlib/LinearAlgebra/src/diagonal.jl
index e8638d3d7ff12..063dde619bd1a 100644
--- a/stdlib/LinearAlgebra/src/diagonal.jl
+++ b/stdlib/LinearAlgebra/src/diagonal.jl
@@ -310,15 +310,6 @@ function (*)(D::Diagonal, V::AbstractVector)
     return D.diag .* V
 end
 
-(*)(A::AbstractMatrix, D::Diagonal) =
-    mul!(similar(A, promote_op(*, eltype(A), eltype(D.diag))), A, D)
-(*)(A::HermOrSym, D::Diagonal) =
-    mul!(similar(A, promote_op(*, eltype(A), eltype(D.diag)), size(A)), A, D)
-(*)(D::Diagonal, A::AbstractMatrix) =
-    mul!(similar(A, promote_op(*, eltype(D.diag), eltype(A))), D, A)
-(*)(D::Diagonal, A::HermOrSym) =
-    mul!(similar(A, promote_op(*, eltype(A), eltype(D.diag)), size(A)), D, A)
-
 rmul!(A::AbstractMatrix, D::Diagonal) = @inline mul!(A, A, D)
 lmul!(D::Diagonal, B::AbstractVecOrMat) = @inline mul!(B, D, B)
 
@@ -431,8 +422,8 @@ function (*)(Da::Diagonal, Db::Diagonal, Dc::Diagonal)
     return Diagonal(Da.diag .* Db.diag .* Dc.diag)
 end
 
-/(A::AbstractVecOrMat, D::Diagonal) = _rdiv!(similar(A, _init_eltype(/, eltype(A), eltype(D))), A, D)
-/(A::HermOrSym, D::Diagonal) = _rdiv!(similar(A, _init_eltype(/, eltype(A), eltype(D)), size(A)), A, D)
+/(A::AbstractVecOrMat, D::Diagonal) = _rdiv!(matprod_dest(A, D, promote_op(/, eltype(A), eltype(D))), A, D)
+/(A::HermOrSym, D::Diagonal) = _rdiv!(matprod_dest(A, D, promote_op(/, eltype(A), eltype(D))), A, D)
 
 rdiv!(A::AbstractVecOrMat, D::Diagonal) = @inline _rdiv!(A, A, D)
 # avoid copy when possible via internal 3-arg backend
@@ -458,8 +449,8 @@ function \(D::Diagonal, B::AbstractVector)
     isnothing(j) || throw(SingularException(j))
     return D.diag .\ B
 end
-\(D::Diagonal, B::AbstractMatrix) = ldiv!(similar(B, _init_eltype(\, eltype(D), eltype(B))), D, B)
-\(D::Diagonal, B::HermOrSym) = ldiv!(similar(B, _init_eltype(\, eltype(D), eltype(B)), size(B)), D, B)
+\(D::Diagonal, B::AbstractMatrix) = ldiv!(matprod_dest(D, B, promote_op(\, eltype(D), eltype(B))), D, B)
+\(D::Diagonal, B::HermOrSym) = ldiv!(matprod_dest(D, B, promote_op(\, eltype(D), eltype(B))), D, B)
 
 ldiv!(D::Diagonal, B::AbstractVecOrMat) = @inline ldiv!(B, D, B)
 function ldiv!(B::AbstractVecOrMat, D::Diagonal, A::AbstractVecOrMat)
@@ -479,8 +470,8 @@ function ldiv!(B::AbstractVecOrMat, D::Diagonal, A::AbstractVecOrMat)
 end
 
 # Optimizations for \, / between Diagonals
-\(D::Diagonal, B::Diagonal) = ldiv!(similar(B, promote_op(\, eltype(D), eltype(B))), D, B)
-/(A::Diagonal, D::Diagonal) = _rdiv!(similar(A, promote_op(/, eltype(A), eltype(D))), A, D)
+\(D::Diagonal, B::Diagonal) = ldiv!(matprod_dest(D, B, promote_op(\, eltype(D), eltype(B))), D, B)
+/(A::Diagonal, D::Diagonal) = _rdiv!(matprod_dest(A, D, promote_op(/, eltype(A), eltype(D))), A, D)
 function _rdiv!(Dc::Diagonal, Db::Diagonal, Da::Diagonal)
     n, k = length(Db.diag), length(Da.diag)
     n == k || throw(DimensionMismatch("left hand side has $n columns but D is $k by $k"))
@@ -543,7 +534,7 @@ function (/)(S::SymTridiagonal, D::Diagonal)
     dl = similar(S.ev, T, max(length(S.dv)-1, 0))
     _rdiv!(Tridiagonal(dl, d, du), S, D)
 end
-(/)(T::Tridiagonal, D::Diagonal) = _rdiv!(similar(T, promote_op(/, eltype(T), eltype(D))), T, D)
+(/)(T::Tridiagonal, D::Diagonal) = _rdiv!(matprod_dest(T, D, promote_op(/, eltype(T), eltype(D))), T, D)
 function _rdiv!(T::Tridiagonal, S::Union{SymTridiagonal,Tridiagonal}, D::Diagonal)
     n = size(S, 2)
     dd = D.diag
@@ -876,9 +867,6 @@ function svd(D::Diagonal{T}) where {T<:Number}
     return SVD(U, S, Vt)
 end
 
-# disambiguation methods: * and / of Diagonal and Adj/Trans AbsVec
-*(u::AdjointAbsVec, D::Diagonal) = (D'u')'
-*(u::TransposeAbsVec, D::Diagonal) = transpose(transpose(D) * transpose(u))
 *(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) = (D' \ u')'
diff --git a/stdlib/LinearAlgebra/src/hessenberg.jl b/stdlib/LinearAlgebra/src/hessenberg.jl
index aed8b3c3747a1..3be41baf24b24 100644
--- a/stdlib/LinearAlgebra/src/hessenberg.jl
+++ b/stdlib/LinearAlgebra/src/hessenberg.jl
@@ -133,29 +133,29 @@ for T = (:Number, :UniformScaling, :Diagonal)
 end
 
 function *(H::UpperHessenberg, U::UpperOrUnitUpperTriangular)
-    HH = mul!(_initarray(*, eltype(H), eltype(U), H), H, U)
+    HH = mul!(matprod_dest(H, U, promote_op(matprod, eltype(H), eltype(U))), H, U)
     UpperHessenberg(HH)
 end
 function *(U::UpperOrUnitUpperTriangular, H::UpperHessenberg)
-    HH = mul!(_initarray(*, eltype(U), eltype(H), H), U, H)
+    HH = mul!(matprod_dest(U, H, promote_op(matprod, eltype(U), eltype(H))), U, H)
     UpperHessenberg(HH)
 end
 
 function /(H::UpperHessenberg, U::UpperTriangular)
-    HH = _rdiv!(_initarray(/, eltype(H), eltype(U), H), H, U)
+    HH = _rdiv!(matprod_dest(H, U, promote_op(/, eltype(H), eltype(U))), H, U)
     UpperHessenberg(HH)
 end
 function /(H::UpperHessenberg, U::UnitUpperTriangular)
-    HH = _rdiv!(_initarray(/, eltype(H), eltype(U), H), H, U)
+    HH = _rdiv!(matprod_dest(H, U, promote_op(/, eltype(H), eltype(U))), H, U)
     UpperHessenberg(HH)
 end
 
 function \(U::UpperTriangular, H::UpperHessenberg)
-    HH = ldiv!(_initarray(\, eltype(U), eltype(H), H), U, H)
+    HH = ldiv!(matprod_dest(U, H, promote_op(\, eltype(U), eltype(H))), U, H)
     UpperHessenberg(HH)
 end
 function \(U::UnitUpperTriangular, H::UpperHessenberg)
-    HH = ldiv!(_initarray(\, eltype(U), eltype(H), H), U, H)
+    HH = ldiv!(matprod_dest(U, H, promote_op(\, eltype(U), eltype(H))), U, H)
     UpperHessenberg(HH)
 end
 
diff --git a/stdlib/LinearAlgebra/src/matmul.jl b/stdlib/LinearAlgebra/src/matmul.jl
index cc1c954258a88..afc646c149eaf 100644
--- a/stdlib/LinearAlgebra/src/matmul.jl
+++ b/stdlib/LinearAlgebra/src/matmul.jl
@@ -106,8 +106,11 @@ julia> [1 1; 0 1] * [1 0; 1 1]
 """
 function (*)(A::AbstractMatrix, B::AbstractMatrix)
     TS = promote_op(matprod, eltype(A), eltype(B))
-    mul!(similar(B, TS, (size(A, 1), size(B, 2))), A, B)
+    mul!(matprod_dest(A, B, TS), A, B)
 end
+
+matprod_dest(A, B, TS) = similar(B, TS, (size(A, 1), size(B, 2)))
+
 # optimization for dispatching to BLAS, e.g. *(::Matrix{Float32}, ::Matrix{Float64})
 # but avoiding the case *(::Matrix{<:BlasComplex}, ::Matrix{<:BlasReal})
 # which is better handled by reinterpreting rather than promotion
diff --git a/stdlib/LinearAlgebra/src/triangular.jl b/stdlib/LinearAlgebra/src/triangular.jl
index 64e1b320f19dd..781e323a0ca43 100644
--- a/stdlib/LinearAlgebra/src/triangular.jl
+++ b/stdlib/LinearAlgebra/src/triangular.jl
@@ -1545,22 +1545,11 @@ rmul!(A::LowerTriangular, B::UnitLowerTriangular) = LowerTriangular(rmul!(tril!(
 ## necessary in the general triangular solve problem.
 
 _inner_type_promotion(op, ::Type{TA}, ::Type{TB}) where {TA<:Integer,TB<:Integer} =
-    _init_eltype(*, TA, TB)
+    promote_op(matprod, TA, TB)
 _inner_type_promotion(op, ::Type{TA}, ::Type{TB}) where {TA,TB} =
-    _init_eltype(op, TA, TB)
+    promote_op(op, TA, TB)
 ## The general promotion methods
-function *(A::AbstractTriangular, B::AbstractTriangular)
-    TAB = _init_eltype(*, eltype(A), eltype(B))
-    mul!(similar(B, TAB, size(B)), A, B)
-end
-
 for mat in (:AbstractVector, :AbstractMatrix)
-    ### Multiplication with triangle to the left and hence rhs cannot be transposed.
-    @eval function *(A::AbstractTriangular, B::$mat)
-        require_one_based_indexing(B)
-        TAB = _init_eltype(*, eltype(A), eltype(B))
-        mul!(similar(B, TAB, size(B)), A, B)
-    end
     ### Left division with triangle to the left hence rhs cannot be transposed. No quotients.
     @eval function \(A::Union{UnitUpperTriangular,UnitLowerTriangular}, B::$mat)
         require_one_based_indexing(B)
@@ -1570,7 +1559,7 @@ for mat in (:AbstractVector, :AbstractMatrix)
     ### Left division with triangle to the left hence rhs cannot be transposed. Quotients.
     @eval function \(A::Union{UpperTriangular,LowerTriangular}, B::$mat)
         require_one_based_indexing(B)
-        TAB = _init_eltype(\, eltype(A), eltype(B))
+        TAB = promote_op(\, eltype(A), eltype(B))
         ldiv!(similar(B, TAB, size(B)), A, B)
     end
     ### Right division with triangle to the right hence lhs cannot be transposed. No quotients.
@@ -1582,20 +1571,10 @@ for mat in (:AbstractVector, :AbstractMatrix)
     ### Right division with triangle to the right hence lhs cannot be transposed. Quotients.
     @eval function /(A::$mat, B::Union{UpperTriangular,LowerTriangular})
         require_one_based_indexing(A)
-        TAB = _init_eltype(/, eltype(A), eltype(B))
+        TAB = promote_op(/, eltype(A), eltype(B))
         _rdiv!(similar(A, TAB, size(A)), A, B)
     end
 end
-### Multiplication with triangle to the right and hence lhs cannot be transposed.
-# Only for AbstractMatrix, hence outside the above loop.
-function *(A::AbstractMatrix, B::AbstractTriangular)
-    require_one_based_indexing(A)
-    TAB = _init_eltype(*, eltype(A), eltype(B))
-    mul!(similar(A, TAB, size(A)), A, B)
-end
-# ambiguity resolution with definitions in matmul.jl
-*(v::AdjointAbsVec, A::AbstractTriangular) = adjoint(adjoint(A) * v.parent)
-*(v::TransposeAbsVec, A::AbstractTriangular) = transpose(transpose(A) * v.parent)
 
 ## Some Triangular-Triangular cases. We might want to write tailored methods
 ## for these cases, but I'm not sure it is worth it.
diff --git a/stdlib/LinearAlgebra/test/diagonal.jl b/stdlib/LinearAlgebra/test/diagonal.jl
index 9db7d8071931c..dab8f45ef66c7 100644
--- a/stdlib/LinearAlgebra/test/diagonal.jl
+++ b/stdlib/LinearAlgebra/test/diagonal.jl
@@ -1242,6 +1242,14 @@ end
     end
 end
 
+@testset "avoid matmul ambiguities with ::MyMatrix * ::AbstractMatrix" begin
+    A = [i+j for i in 1:2, j in 1:2]
+    S = SizedArrays.SizedArray{(2,2)}(A)
+    D = Diagonal([1:2;])
+    @test S * D == A * D
+    @test D * S == D * A
+end
+
 @testset "copy" begin
     @test copy(Diagonal(1:5)) === Diagonal(1:5)
 end
diff --git a/stdlib/LinearAlgebra/test/triangular.jl b/stdlib/LinearAlgebra/test/triangular.jl
index b60efba1d941a..e245240335434 100644
--- a/stdlib/LinearAlgebra/test/triangular.jl
+++ b/stdlib/LinearAlgebra/test/triangular.jl
@@ -896,6 +896,14 @@ end
     end
 end
 
+@testset "avoid matmul ambiguities with ::MyMatrix * ::AbstractMatrix" begin
+    A = [i+j for i in 1:2, j in 1:2]
+    S = SizedArrays.SizedArray{(2,2)}(A)
+    U = UpperTriangular(ones(2,2))
+    @test S * U == A * U
+    @test U * S == U * A
+end
+
 @testset "custom axes" begin
     SZA = SizedArrays.SizedArray{(2,2)}([1 2; 3 4])
     for T in (UpperTriangular, LowerTriangular, UnitUpperTriangular, UnitLowerTriangular)
diff --git a/test/testhelpers/SizedArrays.jl b/test/testhelpers/SizedArrays.jl
index a4810e297e0f9..67f5059b36efd 100644
--- a/test/testhelpers/SizedArrays.jl
+++ b/test/testhelpers/SizedArrays.jl
@@ -56,4 +56,9 @@ function *(S1::SizedArrayLike, S2::SizedArrayLike)
     SZ = ndims(data) == 1 ? (size(S1, 1), ) : (size(S1, 1), size(S2, 2))
     SizedArray{SZ}(data)
 end
+
+# deliberately wide method definition to ensure that this doesn't lead to ambiguities with
+# structured matrices
+*(S1::SizedArrayLike, M::AbstractMatrix) = _data(S1) * M
+
 end