diff --git a/stdlib/LinearAlgebra/src/symmetric.jl b/stdlib/LinearAlgebra/src/symmetric.jl
index 38035e09b4375..b4b5a8825da8c 100644
--- a/stdlib/LinearAlgebra/src/symmetric.jl
+++ b/stdlib/LinearAlgebra/src/symmetric.jl
@@ -269,10 +269,34 @@ end
     end
 end
 
+_conjugation(::Symmetric) = transpose
+_conjugation(::Hermitian) = adjoint
+
 diag(A::Symmetric) = symmetric.(diag(parent(A)), sym_uplo(A.uplo))
 diag(A::Hermitian) = hermitian.(diag(parent(A)), sym_uplo(A.uplo))
 
-isdiag(A::HermOrSym) = isdiag(A.uplo == 'U' ? UpperTriangular(A.data) : LowerTriangular(A.data))
+function applytri(f, A::HermOrSym)
+    if A.uplo == 'U'
+        f(UpperTriangular(A.data))
+    else
+        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))
+    elseif A.uplo == B.uplo == 'L'
+        f(LowerTriangular(A.data), LowerTriangular(B.data))
+    elseif A.uplo == 'U'
+        f(UpperTriangular(A.data), UpperTriangular(_conjugation(B)(B.data)))
+    else # A.uplo == 'L'
+        f(UpperTriangular(_conjugation(A)(A.data)), UpperTriangular(B.data))
+    end
+end
+parentof_applytri(f, args...) = applytri(parent ∘ f, args...)
+
+isdiag(A::HermOrSym) = applytri(isdiag, A)
 
 # For A<:Union{Symmetric,Hermitian}, similar(A[, neweltype]) should yield a matrix with the same
 # symmetry type, uplo flag, and underlying storage type as A. The following methods cover these cases.
@@ -314,8 +338,8 @@ Hermitian{T,S}(A::Hermitian) where {T,S<:AbstractMatrix{T}} = Hermitian{T,S}(con
 AbstractMatrix{T}(A::Hermitian) where {T} = Hermitian(convert(AbstractMatrix{T}, A.data), sym_uplo(A.uplo))
 AbstractMatrix{T}(A::Hermitian{T}) where {T} = copy(A)
 
-copy(A::Symmetric{T,S}) where {T,S} = (B = copy(A.data); Symmetric{T,typeof(B)}(B,A.uplo))
-copy(A::Hermitian{T,S}) where {T,S} = (B = copy(A.data); Hermitian{T,typeof(B)}(B,A.uplo))
+copy(A::Symmetric) = (Symmetric(parentof_applytri(copy, A), sym_uplo(A.uplo)))
+copy(A::Hermitian) = (Hermitian(parentof_applytri(copy, A), sym_uplo(A.uplo)))
 
 function copyto!(dest::Symmetric, src::Symmetric)
     if src.uplo == dest.uplo
@@ -389,9 +413,9 @@ transpose(A::Hermitian) = Transpose(A)
 
 real(A::Symmetric{<:Real}) = A
 real(A::Hermitian{<:Real}) = A
-real(A::Symmetric) = Symmetric(real(A.data), sym_uplo(A.uplo))
-real(A::Hermitian) = Hermitian(real(A.data), sym_uplo(A.uplo))
-imag(A::Symmetric) = Symmetric(imag(A.data), sym_uplo(A.uplo))
+real(A::Symmetric) = Symmetric(parentof_applytri(real, A), sym_uplo(A.uplo))
+real(A::Hermitian) = Hermitian(parentof_applytri(real, A), sym_uplo(A.uplo))
+imag(A::Symmetric) = Symmetric(parentof_applytri(imag, A), sym_uplo(A.uplo))
 
 Base.copy(A::Adjoint{<:Any,<:Symmetric}) =
     Symmetric(copy(adjoint(A.parent.data)), ifelse(A.parent.uplo == 'U', :L, :U))
@@ -401,8 +425,9 @@ Base.copy(A::Transpose{<:Any,<:Hermitian}) =
 tr(A::Symmetric) = tr(A.data) # to avoid AbstractMatrix fallback (incl. allocations)
 tr(A::Hermitian) = real(tr(A.data))
 
-Base.conj(A::HermOrSym) = typeof(A)(conj(A.data), A.uplo)
-Base.conj!(A::HermOrSym) = typeof(A)(conj!(A.data), A.uplo)
+Base.conj(A::Symmetric) = Symmetric(parentof_applytri(conj, A), sym_uplo(A.uplo))
+Base.conj(A::Hermitian) = Hermitian(parentof_applytri(conj, A), sym_uplo(A.uplo))
+Base.conj!(A::HermOrSym) = typeof(A)(parentof_applytri(conj!, A), A.uplo)
 
 # tril/triu
 function tril(A::Hermitian, k::Integer=0)
@@ -496,21 +521,14 @@ for (T, trans, real) in [(:Symmetric, :transpose, :identity), (:(Hermitian{<:Uni
     end
 end
 
-(-)(A::Symmetric) = Symmetric(-A.data, sym_uplo(A.uplo))
-(-)(A::Hermitian) = Hermitian(-A.data, sym_uplo(A.uplo))
+(-)(A::Symmetric) = Symmetric(parentof_applytri(-, A), sym_uplo(A.uplo))
+(-)(A::Hermitian) = Hermitian(parentof_applytri(-, A), sym_uplo(A.uplo))
 
 ## Addition/subtraction
-for f ∈ (:+, :-), (Wrapper, conjugation) ∈ ((:Hermitian, :adjoint), (:Symmetric, :transpose))
-    @eval begin
-        function $f(A::$Wrapper, B::$Wrapper)
-            if A.uplo == B.uplo
-                return $Wrapper($f(parent(A), parent(B)), sym_uplo(A.uplo))
-            elseif A.uplo == 'U'
-                return $Wrapper($f(parent(A), $conjugation(parent(B))), :U)
-            else
-                return $Wrapper($f($conjugation(parent(A)), parent(B)), :U)
-            end
-        end
+for f ∈ (:+, :-), Wrapper ∈ (:Hermitian, :Symmetric)
+    @eval function $f(A::$Wrapper, B::$Wrapper)
+        uplo = A.uplo == B.uplo ? sym_uplo(A.uplo) : (:U)
+        $Wrapper(parentof_applytri($f, A, B), uplo)
     end
 end
 
@@ -555,12 +573,12 @@ function dot(x::AbstractVector, A::RealHermSymComplexHerm, y::AbstractVector)
 end
 
 # Scaling with Number
-*(A::Symmetric, x::Number) = Symmetric(A.data*x, sym_uplo(A.uplo))
-*(x::Number, A::Symmetric) = Symmetric(x*A.data, sym_uplo(A.uplo))
-*(A::Hermitian, x::Real) = Hermitian(A.data*x, sym_uplo(A.uplo))
-*(x::Real, A::Hermitian) = Hermitian(x*A.data, sym_uplo(A.uplo))
-/(A::Symmetric, x::Number) = Symmetric(A.data/x, sym_uplo(A.uplo))
-/(A::Hermitian, x::Real) = Hermitian(A.data/x, sym_uplo(A.uplo))
+*(A::Symmetric, x::Number) = Symmetric(parentof_applytri(y -> y * x, A), sym_uplo(A.uplo))
+*(x::Number, A::Symmetric) = Symmetric(parentof_applytri(y -> x * y, A), sym_uplo(A.uplo))
+*(A::Hermitian, x::Real) = Hermitian(parentof_applytri(y -> y * x, A), sym_uplo(A.uplo))
+*(x::Real, A::Hermitian) = Hermitian(parentof_applytri(y -> x * y, A), sym_uplo(A.uplo))
+/(A::Symmetric, x::Number) = Symmetric(parentof_applytri(y -> y/x, A), sym_uplo(A.uplo))
+/(A::Hermitian, x::Real) = Hermitian(parentof_applytri(y -> y/x, A), sym_uplo(A.uplo))
 
 factorize(A::HermOrSym) = _factorize(A)
 function _factorize(A::HermOrSym{T}; check::Bool=true) where T
diff --git a/stdlib/LinearAlgebra/test/symmetric.jl b/stdlib/LinearAlgebra/test/symmetric.jl
index b3e0b7b560e7a..d3b24ccf78b0b 100644
--- a/stdlib/LinearAlgebra/test/symmetric.jl
+++ b/stdlib/LinearAlgebra/test/symmetric.jl
@@ -470,6 +470,42 @@ end
     end
 end
 
+@testset "non-isbits algebra" begin
+    for ST in (Symmetric, Hermitian), uplo in (:L, :U)
+        M = Matrix{Complex{BigFloat}}(undef,2,2)
+        M[1,1] = rand()
+        M[2,2] = rand()
+        M[1+(uplo==:L), 1+(uplo==:U)] = rand(ComplexF64)
+        S = ST(M, uplo)
+        MS = Matrix(S)
+        @test real(S) == real(MS)
+        @test imag(S) == imag(MS)
+        @test conj(S) == conj(MS)
+        @test conj!(copy(S)) == conj(MS)
+        @test -S == -MS
+        @test S + S == MS + MS
+        @test S - S == MS - MS
+        @test S*2 == 2*S == 2*MS
+        @test S/2 == MS/2
+    end
+    @testset "mixed uplo" begin
+        Mu = Matrix{Complex{BigFloat}}(undef,2,2)
+        Mu[1,1] = Mu[2,2] = 3
+        Mu[1,2] = 2 + 3im
+        Ml = Matrix{Complex{BigFloat}}(undef,2,2)
+        Ml[1,1] = Ml[2,2] = 4
+        Ml[2,1] = 4 + 5im
+        for ST in (Symmetric, Hermitian)
+            Su = ST(Mu, :U)
+            MSu = Matrix(Su)
+            Sl = ST(Ml, :L)
+            MSl = Matrix(Sl)
+            @test Su + Sl == Sl + Su == MSu + MSl
+            @test Su - Sl == -(Sl - Su) == MSu - MSl
+        end
+    end
+end
+
 # bug identified in PR #52318: dot products of quaternionic Hermitian matrices,
 # or any number type where conj(a)*conj(b) ≠ conj(a*b):
 @testset "dot Hermitian quaternion #52318" begin
@@ -932,4 +968,11 @@ end
     end
 end
 
+@testset "conj for immutable" begin
+    S = Symmetric(reshape((1:16)*im, 4, 4))
+    @test conj(S) == conj(Array(S))
+    H = Hermitian(reshape((1:16)*im, 4, 4))
+    @test conj(H) == conj(Array(H))
+end
+
 end # module TestSymmetric