diff --git a/stdlib/LinearAlgebra/src/bidiag.jl b/stdlib/LinearAlgebra/src/bidiag.jl
index 90c91f473e0663..3a3afcb8d5c506 100644
--- a/stdlib/LinearAlgebra/src/bidiag.jl
+++ b/stdlib/LinearAlgebra/src/bidiag.jl
@@ -275,7 +275,9 @@ for func in (:conj, :copy, :real, :imag)
     @eval ($func)(M::Bidiagonal) = Bidiagonal(($func)(M.dv), ($func)(M.ev), M.uplo)
 end
 
-adjoint(B::Bidiagonal{<:Number}) = Bidiagonal(conj(B.dv), conj(B.ev), B.uplo == 'U' ? :L : :U)
+adjoint(B::Bidiagonal{<:Number}) = Bidiagonal(vec(adjoint(B.dv)), vec(adjoint(B.ev)), B.uplo == 'U' ? :L : :U)
+adjoint(B::Bidiagonal{<:Number, <:Base.ReshapedArray{<:Number,1,<:Adjoint}}) =
+    Bidiagonal(adjoint(parent(B.dv)), adjoint(parent(B.ev)), B.uplo == 'U' ? :L : :U)
 transpose(B::Bidiagonal{<:Number}) = Bidiagonal(B.dv, B.ev, B.uplo == 'U' ? :L : :U)
 permutedims(B::Bidiagonal) = Bidiagonal(B.dv, B.ev, B.uplo == 'U' ? 'L' : 'U')
 function permutedims(B::Bidiagonal, perm)
diff --git a/stdlib/LinearAlgebra/src/tridiag.jl b/stdlib/LinearAlgebra/src/tridiag.jl
index b60ae6c82b7d81..0a99a4e96e6ecf 100644
--- a/stdlib/LinearAlgebra/src/tridiag.jl
+++ b/stdlib/LinearAlgebra/src/tridiag.jl
@@ -167,7 +167,10 @@ for func in (:conj, :copy, :real, :imag)
 end
 
 transpose(S::SymTridiagonal) = S
-adjoint(S::SymTridiagonal{<:Real}) = S
+adjoint(S::SymTridiagonal{<:Number}) = SymTridiagonal(vec(adjoint(S.dv)), vec(adjoint(S.ev)))
+adjoint(S::SymTridiagonal{<:Number, <:Base.ReshapedArray{<:Number,1,<:Adjoint}}) =
+    SymTridiagonal(adjoint(parent(S.dv)), adjoint(parent(S.ev)))
+
 permutedims(S::SymTridiagonal) = S
 function permutedims(S::SymTridiagonal, perm)
     Base.checkdims_perm(S, S, perm)
@@ -622,7 +625,9 @@ for func in (:conj, :copy, :real, :imag)
     end
 end
 
-adjoint(S::Tridiagonal{<:Real}) = Tridiagonal(S.du, S.d, S.dl)
+adjoint(S::Tridiagonal{<:Number}) = Tridiagonal(vec(adjoint(S.du)), vec(adjoint(S.d)), vec(adjoint(S.dl)))
+adjoint(S::Tridiagonal{<:Number, <:Base.ReshapedArray{<:Number,1,<:Adjoint}}) =
+    Tridiagonal(adjoint(parent(S.du)), adjoint(parent(S.d)), adjoint(parent(S.dl)))
 transpose(S::Tridiagonal{<:Number}) = Tridiagonal(S.du, S.d, S.dl)
 permutedims(T::Tridiagonal) = Tridiagonal(T.du, T.d, T.dl)
 function permutedims(T::Tridiagonal, perm)
diff --git a/stdlib/LinearAlgebra/test/bidiag.jl b/stdlib/LinearAlgebra/test/bidiag.jl
index 25338d641c58f9..5d33041e78500e 100644
--- a/stdlib/LinearAlgebra/test/bidiag.jl
+++ b/stdlib/LinearAlgebra/test/bidiag.jl
@@ -166,6 +166,9 @@ Random.seed!(1)
 
         @testset for func in (conj, transpose, adjoint)
             @test func(func(T)) == T
+            if func ∈ (transpose, adjoint)
+                @test func(func(T)) === T
+            end
         end
 
         @testset "permutedims(::Bidiagonal)" begin
diff --git a/stdlib/LinearAlgebra/test/tridiag.jl b/stdlib/LinearAlgebra/test/tridiag.jl
index 064e0464701ae7..b9717bba426875 100644
--- a/stdlib/LinearAlgebra/test/tridiag.jl
+++ b/stdlib/LinearAlgebra/test/tridiag.jl
@@ -282,6 +282,9 @@ end
         @testset "Idempotent tests" begin
             for func in (conj, transpose, adjoint)
                 @test func(func(A)) == A
+                if func ∈ (transpose, adjoint)
+                    @test func(func(A)) === A
+                end
             end
         end
         @testset "permutedims(::[Sym]Tridiagonal)" begin