diff --git a/stdlib/LinearAlgebra/src/diagonal.jl b/stdlib/LinearAlgebra/src/diagonal.jl
index 32e2fa61fa680..bbeb2a50a9a93 100644
--- a/stdlib/LinearAlgebra/src/diagonal.jl
+++ b/stdlib/LinearAlgebra/src/diagonal.jl
@@ -947,3 +947,6 @@ end
 function Base.muladd(A::Diagonal, B::Diagonal, z::Diagonal)
     Diagonal(A.diag .* B.diag .+ z.diag)
 end
+
+uppertriangular(D::Diagonal) = D
+lowertriangular(D::Diagonal) = D
diff --git a/stdlib/LinearAlgebra/src/symmetric.jl b/stdlib/LinearAlgebra/src/symmetric.jl
index 410140bf7e8be..630cbe78ba5dc 100644
--- a/stdlib/LinearAlgebra/src/symmetric.jl
+++ b/stdlib/LinearAlgebra/src/symmetric.jl
@@ -281,21 +281,21 @@ diag(A::Hermitian) = hermitian.(diag(parent(A)), sym_uplo(A.uplo))
 
 function applytri(f, A::HermOrSym)
     if A.uplo == 'U'
-        f(UpperTriangular(A.data))
+        f(uppertriangular(A.data))
     else
-        f(LowerTriangular(A.data))
+        f(lowertriangular(A.data))
     end
 end
 
 function applytri(f, A::HermOrSym, B::HermOrSym)
     if A.uplo == B.uplo == 'U'
-        f(UpperTriangular(A.data), UpperTriangular(B.data))
+        f(uppertriangular(A.data), uppertriangular(B.data))
     elseif A.uplo == B.uplo == 'L'
-        f(LowerTriangular(A.data), LowerTriangular(B.data))
+        f(lowertriangular(A.data), lowertriangular(B.data))
     elseif A.uplo == 'U'
-        f(UpperTriangular(A.data), UpperTriangular(_conjugation(B)(B.data)))
+        f(uppertriangular(A.data), uppertriangular(_conjugation(B)(B.data)))
     else # A.uplo == 'L'
-        f(UpperTriangular(_conjugation(A)(A.data)), UpperTriangular(B.data))
+        f(uppertriangular(_conjugation(A)(A.data)), uppertriangular(B.data))
     end
 end
 parentof_applytri(f, args...) = applytri(parent ∘ f, args...)
diff --git a/stdlib/LinearAlgebra/src/triangular.jl b/stdlib/LinearAlgebra/src/triangular.jl
index 5c3f728eeaa5e..922bd9a6bd91a 100644
--- a/stdlib/LinearAlgebra/src/triangular.jl
+++ b/stdlib/LinearAlgebra/src/triangular.jl
@@ -154,6 +154,12 @@ const UpperOrUnitUpperTriangular{T,S} = Union{UpperTriangular{T,S}, UnitUpperTri
 const LowerOrUnitLowerTriangular{T,S} = Union{LowerTriangular{T,S}, UnitLowerTriangular{T,S}}
 const UpperOrLowerTriangular{T,S} = Union{UpperOrUnitUpperTriangular{T,S}, LowerOrUnitLowerTriangular{T,S}}
 
+uppertriangular(M) = UpperTriangular(M)
+lowertriangular(M) = LowerTriangular(M)
+
+uppertriangular(U::UpperOrUnitUpperTriangular) = U
+lowertriangular(U::LowerOrUnitLowerTriangular) = U
+
 Base.dataids(A::UpperOrLowerTriangular) = Base.dataids(A.data)
 
 imag(A::UpperTriangular) = UpperTriangular(imag(A.data))
diff --git a/stdlib/LinearAlgebra/test/diagonal.jl b/stdlib/LinearAlgebra/test/diagonal.jl
index aa960aaa53193..91e69c0e30bf9 100644
--- a/stdlib/LinearAlgebra/test/diagonal.jl
+++ b/stdlib/LinearAlgebra/test/diagonal.jl
@@ -1281,6 +1281,12 @@ end
     @test c == Diagonal([2,2,2,2])
 end
 
+@testset "uppertriangular/lowertriangular" begin
+    D = Diagonal([1,2])
+    @test LinearAlgebra.uppertriangular(D) === D
+    @test LinearAlgebra.lowertriangular(D) === D
+end
+
 @testset "mul/div with an adjoint vector" begin
     A = [1.0;;]
     x = [1.0]
diff --git a/stdlib/LinearAlgebra/test/symmetric.jl b/stdlib/LinearAlgebra/test/symmetric.jl
index 7b42825a12681..f9da7d004822b 100644
--- a/stdlib/LinearAlgebra/test/symmetric.jl
+++ b/stdlib/LinearAlgebra/test/symmetric.jl
@@ -532,6 +532,31 @@ end
             @test kron(Sl,Su) == kron(MSl,MSu)
         end
     end
+    @testset "non-strided" begin
+        @testset "diagonal" begin
+            for ST1 in (Symmetric, Hermitian), uplo1 in (:L, :U)
+                m = ST1(Matrix{BigFloat}(undef,2,2), uplo1)
+                m.data[1,1] = 1
+                m.data[2,2] = 3
+                m.data[1+(uplo1==:L), 1+(uplo1==:U)] = 2
+                A = Array(m)
+                for ST2 in (Symmetric, Hermitian), uplo2 in (:L, :U)
+                    id = ST2(I(2), uplo2)
+                    @test m + id == id + m == A + id
+                end
+            end
+        end
+        @testset "unit triangular" begin
+            for ST1 in (Symmetric, Hermitian), uplo1 in (:L, :U)
+                H1 = ST1(UnitUpperTriangular(big.(rand(Int8,4,4))), uplo1)
+                M1 = Matrix(H1)
+                for ST2 in (Symmetric, Hermitian), uplo2 in (:L, :U)
+                    H2 = ST2(UnitUpperTriangular(big.(rand(Int8,4,4))), uplo2)
+                    @test H1 + H2 == M1 + Matrix(H2)
+                end
+            end
+        end
+    end
 end
 
 # bug identified in PR #52318: dot products of quaternionic Hermitian matrices,
diff --git a/stdlib/LinearAlgebra/test/triangular.jl b/stdlib/LinearAlgebra/test/triangular.jl
index 9c23ec92fdc74..40319b644b3cf 100644
--- a/stdlib/LinearAlgebra/test/triangular.jl
+++ b/stdlib/LinearAlgebra/test/triangular.jl
@@ -998,6 +998,14 @@ end
     end
 end
 
+@testset "uppertriangular/lowertriangular" begin
+    M = rand(2,2)
+    @test LinearAlgebra.uppertriangular(M) === UpperTriangular(M)
+    @test LinearAlgebra.lowertriangular(M) === LowerTriangular(M)
+    @test LinearAlgebra.uppertriangular(UnitUpperTriangular(M)) === UnitUpperTriangular(M)
+    @test LinearAlgebra.lowertriangular(UnitLowerTriangular(M)) === UnitLowerTriangular(M)
+end
+
 @testset "arithmetic with partly uninitialized matrices" begin
     @testset "$(typeof(A))" for A in (Matrix{BigFloat}(undef,2,2), Matrix{Complex{BigFloat}}(undef,2,2)')
         A[2,1] = eltype(A) <: Complex ? 4 + 3im : 4