diff --git a/stdlib/LinearAlgebra/src/hessenberg.jl b/stdlib/LinearAlgebra/src/hessenberg.jl
index 8502ef896a68b..efdab9b9d8542 100644
--- a/stdlib/LinearAlgebra/src/hessenberg.jl
+++ b/stdlib/LinearAlgebra/src/hessenberg.jl
@@ -67,7 +67,7 @@ true
 hessenberg(A::StridedMatrix{T}) where T =
     hessenberg!(copy_oftype(A, eigtype(T)))
 
-struct HessenbergQ{T,S<:AbstractMatrix} <: AbstractMatrix{T}
+struct HessenbergQ{T,S<:AbstractMatrix} <: AbstractQ{T}
     factors::S
     τ::Vector{T}
     function HessenbergQ{T,S}(factors, τ) where {T,S<:AbstractMatrix}
@@ -77,8 +77,6 @@ struct HessenbergQ{T,S<:AbstractMatrix} <: AbstractMatrix{T}
 end
 HessenbergQ(factors::AbstractMatrix{T}, τ::Vector{T}) where {T} = HessenbergQ{T,typeof(factors)}(factors, τ)
 HessenbergQ(A::Hessenberg) = HessenbergQ(A.factors, A.τ)
-size(A::HessenbergQ, d) = size(A.factors, d)
-size(A::HessenbergQ) = size(A.factors)
 
 function getproperty(F::Hessenberg, d::Symbol)
     d == :Q && return HessenbergQ(F)
@@ -89,17 +87,8 @@ end
 Base.propertynames(F::Hessenberg, private::Bool=false) =
     (:Q, :H, (private ? fieldnames(typeof(F)) : ())...)
 
-function getindex(A::HessenbergQ, i::Integer, j::Integer)
-    x = zeros(eltype(A), size(A, 1))
-    x[i] = 1
-    y = zeros(eltype(A), size(A, 2))
-    y[j] = 1
-    return dot(x, lmul!(A, y))
-end
-
 ## reconstruct the original matrix
-Matrix(A::HessenbergQ{<:BlasFloat}) = LAPACK.orghr!(1, size(A.factors, 1), copy(A.factors), A.τ)
-Array(A::HessenbergQ) = Matrix(A)
+Matrix{T}(Q::HessenbergQ) where {T} = convert(Matrix{T}, LAPACK.orghr!(1, size(Q.factors, 1), copy(Q.factors), Q.τ))
 AbstractMatrix(F::Hessenberg) = (fq = Array(F.Q); (fq * F.H) * fq')
 AbstractArray(F::Hessenberg) = AbstractMatrix(F)
 Matrix(F::Hessenberg) = Array(AbstractArray(F))
@@ -113,22 +102,3 @@ lmul!(adjQ::Adjoint{<:Any,<:HessenbergQ{T}}, X::StridedVecOrMat{T}) where {T<:Bl
     (Q = adjQ.parent; LAPACK.ormhr!('L', ifelse(T<:Real, 'T', 'C'), 1, size(Q.factors, 1), Q.factors, Q.τ, X))
 rmul!(X::StridedMatrix{T}, adjQ::Adjoint{<:Any,<:HessenbergQ{T}}) where {T<:BlasFloat} =
     (Q = adjQ.parent; LAPACK.ormhr!('R', ifelse(T<:Real, 'T', 'C'), 1, size(Q.factors, 1), Q.factors, Q.τ, X))
-
-function (*)(Q::HessenbergQ{T}, X::StridedVecOrMat{S}) where {T,S}
-    TT = typeof(zero(T)*zero(S) + zero(T)*zero(S))
-    return lmul!(Q, copy_oftype(X, TT))
-end
-function (*)(X::StridedVecOrMat{S}, Q::HessenbergQ{T}) where {T,S}
-    TT = typeof(zero(T)*zero(S) + zero(T)*zero(S))
-    return rmul!(copy_oftype(X, TT), Q)
-end
-function *(adjQ::Adjoint{<:Any,<:HessenbergQ{T}}, X::StridedVecOrMat{S}) where {T,S}
-    Q = adjQ.parent
-    TT = typeof(zero(T)*zero(S) + zero(T)*zero(S))
-    return lmul!(adjoint(Q), copy_oftype(X, TT))
-end
-function *(X::StridedVecOrMat{S}, adjQ::Adjoint{<:Any,<:HessenbergQ{T}}) where {T,S}
-    Q = adjQ.parent
-    TT = typeof(zero(T)*zero(S) + zero(T)*zero(S))
-    return rmul!(copy_oftype(X, TT), adjoint(Q))
-end
diff --git a/stdlib/LinearAlgebra/src/lq.jl b/stdlib/LinearAlgebra/src/lq.jl
index 1cc1b746e349f..a5629335c5e88 100644
--- a/stdlib/LinearAlgebra/src/lq.jl
+++ b/stdlib/LinearAlgebra/src/lq.jl
@@ -113,7 +113,9 @@ end
 
 LQPackedQ{T}(Q::LQPackedQ) where {T} = LQPackedQ(convert(AbstractMatrix{T}, Q.factors), convert(Vector{T}, Q.τ))
 AbstractMatrix{T}(Q::LQPackedQ) where {T} = LQPackedQ{T}(Q)
-Matrix(A::LQPackedQ) = LAPACK.orglq!(copy(A.factors),A.τ)
+Matrix{T}(A::LQPackedQ) where {T} = convert(Matrix{T}, LAPACK.orglq!(copy(A.factors),A.τ))
+Matrix(A::LQPackedQ{T}) where {T} = Matrix{T}(A)
+Array{T}(A::LQPackedQ{T}) where {T} = Matrix{T}(A)
 Array(A::LQPackedQ) = Matrix(A)
 
 size(F::LQ, dim::Integer) = size(getfield(F, :factors), dim)
diff --git a/stdlib/LinearAlgebra/src/qr.jl b/stdlib/LinearAlgebra/src/qr.jl
index dab6bd6a88912..d3646af06ac92 100644
--- a/stdlib/LinearAlgebra/src/qr.jl
+++ b/stdlib/LinearAlgebra/src/qr.jl
@@ -500,7 +500,9 @@ AbstractMatrix{T}(Q::QRPackedQ) where {T} = QRPackedQ{T}(Q)
 QRCompactWYQ{S}(Q::QRCompactWYQ) where {S} = QRCompactWYQ(convert(AbstractMatrix{S}, Q.factors), convert(AbstractMatrix{S}, Q.T))
 AbstractMatrix{S}(Q::QRCompactWYQ{S}) where {S} = Q
 AbstractMatrix{S}(Q::QRCompactWYQ) where {S} = QRCompactWYQ{S}(Q)
-Matrix(A::AbstractQ{T}) where {T} = lmul!(A, Matrix{T}(I, size(A.factors, 1), min(size(A.factors)...)))
+Matrix{T}(A::AbstractQ) where {T} = lmul!(A, Matrix{T}(I, size(A.factors, 1), min(size(A.factors)...)))
+Matrix(A::AbstractQ{T}) where {T} = Matrix{T}(A)
+Array{T}(A::AbstractQ) where {T} = Matrix{T}(A)
 Array(A::AbstractQ) = Matrix(A)
 
 size(A::Union{QR,QRCompactWY,QRPivoted}, dim::Integer) = size(getfield(A, :factors), dim)