diff --git a/src/MatrixFactorizations.jl b/src/MatrixFactorizations.jl
index c90b621..04aebe1 100644
--- a/src/MatrixFactorizations.jl
+++ b/src/MatrixFactorizations.jl
@@ -1,6 +1,6 @@
 module MatrixFactorizations
 using Base, LinearAlgebra, ArrayLayouts
-import Base: axes, axes1, getproperty, iterate, tail, oneto
+import Base: axes, axes1, getproperty, iterate, tail, oneto, BroadcastStyle
 import LinearAlgebra: BlasInt, BlasReal, BlasFloat, BlasComplex, axpy!,
    copy_oftype, checksquare, adjoint, transpose, AdjOrTrans, HermOrSym,
    det, logdet, logabsdet, isposdef
@@ -50,12 +50,18 @@ const TransposeFact = isdefined(LinearAlgebra, :TransposeFactorization) ? Linear
 abstract type LayoutQ{T} <: AbstractQ{T} end
 
 
+# Support orthogonal matrices as an abstract matrix
+abstract type LayoutQMatrix{T} <: LayoutMatrix{T} end
+
+const LayoutQTypes{T} = Union{LayoutQ{T}, LayoutQMatrix{T}}
+
+
 @_layoutlmul LayoutQ
 @_layoutlmul AdjointQtype{<:Any,<:LayoutQ}
 @_layoutrmul LayoutQ
 @_layoutrmul AdjointQtype{<:Any,<:LayoutQ}
 
-LinearAlgebra.copymutable(Q::LayoutQ) = copymutable_size(size(Q), Q)
+LinearAlgebra.copymutable(Q::LayoutQTypes) = copymutable_size(size(Q), Q)
 copymutable_size(sz, Q) = lmul!(Q, Matrix{eltype(Q)}(I, sz))
 
 (*)(Q::LayoutQ, b::AbstractVector) = _mul(Q, b)
@@ -106,16 +112,17 @@ end
 *(A::AbstractTriangular, B::LayoutQ) = mul(A, B)
 *(A::AbstractTriangular, B::AdjointQtype{<:Any,<:LayoutQ}) = mul(A, B)
 
-axes(Q::LayoutQ, dim::Integer) = axes(getfield(Q, :factors), dim == 2 ? 1 : dim)
-axes(Q::LayoutQ) = axes(Q, 1), axes(Q, 2)
-copy(Q::LayoutQ) = Q
+axes(Q::LayoutQTypes, dim::Integer) = axes(getfield(Q, :factors), dim == 2 ? 1 : dim)
+axes(Q::LayoutQTypes) = axes(Q, 1), axes(Q, 2)
+copy(Q::LayoutQTypes) = Q
 Base.@propagate_inbounds getindex(A::LayoutQ, I...) = layout_getindex(A, I...)
+Base.@propagate_inbounds getindex(A::LayoutQMatrix, k::Int, j::Int) = AbstractQ(A)[k, j]
 # by default, fall back to AbstractQ  methods
 layout_getindex(A::LayoutQ, I...) =
     Base.invoke(Base.getindex, Tuple{AbstractQ, typeof.(I)...}, A, I...)
 
-size(Q::LayoutQ, dim::Integer) = size(getfield(Q, :factors), dim == 2 ? 1 : dim)
-size(Q::LayoutQ) = size(Q, 1), size(Q, 2)
+size(Q::LayoutQTypes, dim::Integer) = size(getfield(Q, :factors), dim == 2 ? 1 : dim)
+size(Q::LayoutQTypes) = size(Q, 1), size(Q, 2)
 
 include("ul.jl")
 include("qr.jl")
diff --git a/src/qr.jl b/src/qr.jl
index 397c9af..650c8cc 100644
--- a/src/qr.jl
+++ b/src/qr.jl
@@ -145,6 +145,8 @@ ldiv!(F::QR, B::AbstractVecOrMat) = ArrayLayouts.ldiv!(F, B)
 ldiv!(F::QR, B::LayoutVector) = ArrayLayouts.ldiv!(F, B)
 ldiv!(F::QR, B::LayoutMatrix) = ArrayLayouts.ldiv!(F, B)
 
+size(F::QR, dim::Integer) = size(getfield(F, :factors), dim)
+size(F::QR) = size(getfield(F, :factors))
 
 """
     QRPackedQ <: LinearAlgebra.AbstractQ
@@ -163,28 +165,51 @@ end
 
 
 
-QRPackedQ(factors::AbstractMatrix{T}, τ::AbstractVector{T}) where {T} = QRPackedQ{T,typeof(factors),typeof(τ)}(factors, τ)
-function QRPackedQ{T}(factors::AbstractMatrix, τ::AbstractVector) where {T}
-    QRPackedQ(convert(AbstractMatrix{T}, factors), convert(AbstractVector{T}, τ))
+"""
+    QRPackedQMatrix <: LayoutQMatrix
+
+The orthogonal/unitary ``Q`` matrix of a QR factorization stored in [`QR`](@ref),
+conforming to the `AbstractMatrix` interface.
+"""
+struct QRPackedQMatrix{T,S<:AbstractMatrix{T},Tau<:AbstractVector{T}} <: LayoutQMatrix{T}
+    factors::S
+    τ::Tau
+
+    function QRPackedQMatrix{T,S,Tau}(factors, τ) where {T,S<:AbstractMatrix{T},Tau<:AbstractVector{T}}
+        require_one_based_indexing(factors)
+        new{T,S,Tau}(factors, τ)
+    end
+end
+
+
+for QTyp in (:QRPackedQ, :QRPackedQMatrix)
+    @eval begin
+        $QTyp(factors::AbstractMatrix{T}, τ::AbstractVector{T}) where {T} = $QTyp{T,typeof(factors),typeof(τ)}(factors, τ)
+        function $QTyp{T}(factors::AbstractMatrix, τ::AbstractVector) where {T}
+            $QTyp(convert(AbstractMatrix{T}, factors), convert(AbstractVector{T}, τ))
+        end
+
+        $QTyp{T}(Q::$QTyp) where {T} = $QTyp(convert(AbstractMatrix{T}, Q.factors), convert(AbstractVector{T}, Q.τ))
+    end
 end
 
-QRPackedQ{T}(Q::QRPackedQ) where {T} = QRPackedQ(convert(AbstractMatrix{T}, Q.factors), convert(AbstractVector{T}, Q.τ))
 QRPackedQ(Q::LinearAlgebra.QRPackedQ) = QRPackedQ(Q.factors, Q.τ)
 
+const QRPackedQTypes{T,S<:AbstractMatrix{T},Tau<:AbstractVector{T}} = Union{QRPackedQ{T,S,Tau}, QRPackedQMatrix{T,S,Tau}}
+
+Matrix{T}(Q::QRPackedQTypes{S}) where {T,S} = convert(Matrix{T}, lmul!(Q, Matrix{S}(I, size(Q, 1), min(size(Q.factors)...))))
+Matrix(Q::QRPackedQTypes{S}) where {S} = Matrix{S}(Q)
+
 AbstractQ{T}(Q::QRPackedQ{T}) where {T} = Q
 AbstractQ{T}(Q::QRPackedQ) where {T} = QRPackedQ{T}(Q)
+AbstractQ{T}(Q::QRPackedQMatrix{T}) where {T} = QRPackedQ(Q.factors, Q.τ)
+AbstractQ{T}(Q::QRPackedQMatrix) where {T} = QRPackedQ{T}(Q.factors, Q.τ)
+AbstractQ(Q::QRPackedQMatrix) = QRPackedQ(Q.factors, Q.τ)
 convert(::Type{AbstractQ{T}}, Q::QRPackedQ) where {T} = QRPackedQ{T}(Q)
-
-Matrix{T}(Q::QRPackedQ{S}) where {T,S} =
-    convert(Matrix{T}, lmul!(Q, Matrix{S}(I, size(Q, 1), min(size(Q.factors)...))))
-Matrix(Q::QRPackedQ{S}) where {S} = Matrix{S}(Q)
-
 AbstractMatrix{T}(Q::QRPackedQ) where {T} = Matrix{T}(Q)
 
-size(F::QR, dim::Integer) = size(getfield(F, :factors), dim)
-size(F::QR) = size(getfield(F, :factors))
 
-MemoryLayout(::Type{<:QRPackedQ{<:Any,S,T}}) where {S,T} =
+MemoryLayout(::Type{<:QRPackedQTypes{<:Any,S,T}}) where {S,T} =
     QRPackedQLayout{typeof(MemoryLayout(S)),typeof(MemoryLayout(T))}()
 
 MemoryLayout(::Type{<:QR{<:Any,S,T}}) where {S,T} =
@@ -227,3 +252,9 @@ function (\)(A::QR{T}, BIn::VecOrMat{Complex{T}}) where T<:BlasReal
     XX = reshape(collect(reinterpret(Complex{T}, copy(transpose(reshape(X, div(length(X), 2), 2))))), _ret_size(A, BIn))
     return _cut_B(XX, 1:n)
 end
+
+
+
+# support lazy broadcasting
+
+BroadcastStyle(::Type{<:QRPackedQMatrix{<:Any,F}}) where {F} = BroadcastStyle(F) # TODO: broken for banded?
\ No newline at end of file