diff --git a/stdlib/LinearAlgebra/src/abstractq.jl b/stdlib/LinearAlgebra/src/abstractq.jl
index 853286f34048f..334a9a9235972 100644
--- a/stdlib/LinearAlgebra/src/abstractq.jl
+++ b/stdlib/LinearAlgebra/src/abstractq.jl
@@ -22,9 +22,14 @@ promote_rule(::Type{<:AbstractMatrix{T}}, ::Type{<:AbstractQ{T}}) where {T} =
     (@inline; Union{AbstractMatrix{T},AbstractQ{T}})
 
 # conversion
-AbstractQ{S}(Q::AbstractQ{S}) where {S} = Q
-# the following eltype promotion needs to be defined for each subtype
+# the following eltype promotion should be defined for each subtype `QType`
 # convert(::Type{AbstractQ{T}}, Q::QType) where {T} = QType{T}(Q)
+# and then care has to be taken that
+# QType{T}(Q::QType{T}) where T = ...
+# is implemented as a no-op
+
+# the following conversion method ensures functionality when the above method is not defined
+# (as for HessenbergQ), but no eltype conversion is required either (say, in multiplication)
 convert(::Type{AbstractQ{T}}, Q::AbstractQ{T}) where {T} = Q
 convert(::Type{AbstractQ{T}}, adjQ::AdjointQ{T}) where {T} = adjQ
 convert(::Type{AbstractQ{T}}, adjQ::AdjointQ) where {T} = convert(AbstractQ{T}, adjQ.Q)'
@@ -36,7 +41,6 @@ Matrix(Q::AbstractQ{T}) where {T} = Matrix{T}(Q)
 Array{T}(Q::AbstractQ) where {T} = Matrix{T}(Q)
 Array(Q::AbstractQ) = Matrix(Q)
 convert(::Type{T}, Q::AbstractQ) where {T<:Array} = T(Q)
-convert(::Type{T}, Q::AbstractQ) where {T<:Matrix} = T(Q)
 # legacy
 @deprecate(convert(::Type{AbstractMatrix{T}}, Q::AbstractQ) where {T},
     convert(LinearAlgebra.AbstractQ{T}, Q))
@@ -227,11 +231,9 @@ QRCompactWYQ{S}(factors::AbstractMatrix, T::AbstractMatrix) where {S} =
 @deprecate(QRCompactWYQ{S,M}(factors::AbstractMatrix{S}, T::AbstractMatrix{S}) where {S,M},
            QRCompactWYQ{S,M,typeof(T)}(factors, T), false)
 
-QRPackedQ{T}(Q::QRPackedQ) where {T} = QRPackedQ(convert(AbstractMatrix{T}, Q.factors), convert(Vector{T}, Q.τ))
+QRPackedQ{T}(Q::QRPackedQ) where {T} = QRPackedQ(convert(AbstractMatrix{T}, Q.factors), convert(AbstractVector{T}, Q.τ))
 QRCompactWYQ{S}(Q::QRCompactWYQ) where {S} = QRCompactWYQ(convert(AbstractMatrix{S}, Q.factors), convert(AbstractMatrix{S}, Q.T))
 
-AbstractQ{S}(Q::QRPackedQ) where {S} = QRPackedQ{S}(Q)
-AbstractQ{S}(Q::QRCompactWYQ) where {S} = QRCompactWYQ{S}(Q)
 # override generic square fallback
 Matrix{T}(Q::Union{QRCompactWYQ{S},QRPackedQ{S}}) where {T,S} =
     convert(Matrix{T}, lmul!(Q, Matrix{S}(I, size(Q, 1), min(size(Q.factors)...))))
@@ -505,7 +507,7 @@ struct LQPackedQ{T,S<:AbstractMatrix{T},C<:AbstractVector{T}} <: AbstractQ{T}
     τ::C
 end
 
-LQPackedQ{T}(Q::LQPackedQ) where {T} = LQPackedQ(convert(AbstractMatrix{T}, Q.factors), convert(Vector{T}, Q.τ))
+LQPackedQ{T}(Q::LQPackedQ) where {T} = LQPackedQ(convert(AbstractMatrix{T}, Q.factors), convert(AbstractVector{T}, Q.τ))
 @deprecate(AbstractMatrix{T}(Q::LQPackedQ) where {T},
     convert(AbstractQ{T}, Q),
     false)
diff --git a/stdlib/LinearAlgebra/test/abstractq.jl b/stdlib/LinearAlgebra/test/abstractq.jl
index b3ef5a1952244..e3f48c7b2e3fd 100644
--- a/stdlib/LinearAlgebra/test/abstractq.jl
+++ b/stdlib/LinearAlgebra/test/abstractq.jl
@@ -29,6 +29,9 @@ n = 5
         A = rand(T, n, n)
         F = qr(A)
         Q = MyQ(F.Q)
+        @test ndims(Q) == 2
+        T <: Real && @test transpose(Q) == adjoint(Q)
+        T <: Complex && @test_throws ErrorException transpose(Q)
         @test convert(AbstractQ{complex(T)}, Q) isa MyQ{complex(T)}
         @test convert(AbstractQ{complex(T)}, Q') isa AdjointQ{<:complex(T),<:MyQ{complex(T)}}
         @test Q*I ≈ Q.Q*I rtol=2eps(real(T))
@@ -50,14 +53,15 @@ n = 5
             @test mul!(X, transQ(Q), transY(Y)) ≈ transQ(Q) * transY(Y) ≈ transQ(Q.Q) * transY(Y)
             @test mul!(X, transY(Y), transQ(Q)) ≈ transY(Y) * transQ(Q) ≈ transY(Y) * transQ(Q.Q)
         end
-        @test Matrix(Q) ≈ Q[:,:] ≈ copyto!(zeros(T, size(Q)), Q) ≈ Q.Q*I
-        @test Matrix(Q') ≈ (Q')[:,:] ≈ copyto!(zeros(T, size(Q)), Q') ≈ Q.Q'*I
-        @test Q[1,:] == Q.Q[1,:]
-        @test Q[:,1] == Q.Q[:,1]
+        @test convert(Matrix, Q) ≈ Matrix(Q) ≈ Q[:,:] ≈ copyto!(zeros(T, size(Q)), Q) ≈ Q.Q*I
+        @test convert(Matrix, Q') ≈ Matrix(Q') ≈ (Q')[:,:] ≈ copyto!(zeros(T, size(Q)), Q') ≈ Q.Q'*I
+        @test Q[1,:] == Q.Q[1,:] == view(Q, 1, :)
+        @test Q[:,1] == Q.Q[:,1] == view(Q, :, 1)
         @test Q[1,1] == Q.Q[1,1]
         @test Q[:] == Q.Q[:]
-        @test Q[:,1:3] == Q.Q[:,1:3]
+        @test Q[:,1:3] == Q.Q[:,1:3] == view(Q, :, 1:3)
         @test Q[:,1:3] ≈ Matrix(Q)[:,1:3]
+        @test Q[2:3,2:3] == view(Q, 2:3, 2:3) ≈ Matrix(Q)[2:3,2:3]
         @test_throws BoundsError Q[0,1]
         @test_throws BoundsError Q[n+1,1]
         @test_throws BoundsError Q[1,0]