Add array conversions for AbstractQ

[dlfivefifty]

I think all of these should work:

julia> Q = qr(randn(10,5)).Q;

julia> Matrix(Q); # fine

julia> Array(Q); # fine

julia> AbstractMatrix(Q);
ERROR: MethodError: no method matching (AbstractMatrix)(::LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}})
The type `AbstractMatrix` exists, but no method is defined for this combination of argument types when trying to construct it.

Closest candidates are:
  (AbstractMatrix)(::Hessenberg)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/hessenberg.jl:497
  (AbstractMatrix)(::QRPivoted)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/qr.jl:454
  (AbstractMatrix)(::Eigen)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:674
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[9]:1

julia> convert(AbstractMatrix, Q);
ERROR: MethodError: no method matching (AbstractMatrix)(::LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}})
The type `AbstractMatrix` exists, but no method is defined for this combination of argument types when trying to construct it.

Closest candidates are:
  (AbstractMatrix)(::Hessenberg)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/hessenberg.jl:497
  (AbstractMatrix)(::QRPivoted)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/qr.jl:454
  (AbstractMatrix)(::Eigen)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:674
  ...

Stacktrace:
 [1] convert(::Type{AbstractMatrix}, Q::LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}})
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/abstractq.jl:49
 [2] top-level scope
   @ REPL[10]:1

julia> AbstractArray(Q);
ERROR: MethodError: no method matching AbstractArray(::LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}})
The type `AbstractArray` exists, but no method is defined for this combination of argument types when trying to construct it.

Closest candidates are:
  AbstractArray(::Hessenberg)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/hessenberg.jl:494
  AbstractArray(::QRPivoted)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/qr.jl:455
  AbstractArray(::Eigen)
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/eigen.jl:675
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[12]:1

julia> AbstractMatrix{Float64}(Q);
ERROR: MethodError: no method matching AbstractMatrix{Float64}(::LinearAlgebra.QRCompactWYQ{Float64, Matrix{Float64}, Matrix{Float64}})
The type `AbstractMatrix{Float64}` exists, but no method is defined for this combination of argument types when trying to construct it.

Closest candidates are:
  AbstractMatrix{T}(::Bidiagonal{T, V} where V<:AbstractVector{T}) where T
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/bidiag.jl:232
  AbstractMatrix{T}(::Bidiagonal) where T
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/bidiag.jl:231
  AbstractMatrix{T}(::Tridiagonal{T, V} where V<:AbstractVector{T}) where T
   @ LinearAlgebra ~/Projects/julia-1.11/usr/share/julia/stdlib/v1.11/LinearAlgebra/src/tridiag.jl:803
  ...

Stacktrace:
 [1] top-level scope
   @ REPL[13]:1

[dkarrasch]

The conversion is somewhat "special": we currently have

@deprecate(convert(::Type{AbstractMatrix{T}}, Q::AbstractQ) where {T}, convert(LinearAlgebra.AbstractQ{T}, Q))

to mimick older behavior, when the (assumed) intention was to change the eltype, but not the outer QRCompactWYQ type, for instance. We already have

convert(::Type{T}, Q::AbstractQ) where {T<:AbstractArray} = T(Q)

which does create an array in all other cases. Before v1.10, we had convert(AbstractMatrix, Q) return Q itself, i.e., a no-op.

[dlfivefifty]

If you don't want to support convert we can still support AbstractMatrix(::AbstractQ), which just defaults to calling the existing Matrix constructor.

I want it for converting an infinite-dimensional AbstractQ corresponding to a QR of an ∞ x n matrix to an ∞ x n subtype of AbstractMatrix. It can't be a Matrix.

[dkarrasch]

That means you would also need a

(::Type{T})(Q::AbstractQ) where {T<:AbstractMatrix} = T(Q)

method, right? To be able to call InfiniteMatrix(Q) by fallbacks?

[dlfivefifty]

I just wanted some function to turn an AbstractQ to an AbstractMatrix representing the rectangular reduced QR. AbstractMatrix(::AbstractQ) would suffice.

But perhaps the other constructor is useful? One can imagine calling Diagonal(::AbstractQ), especially when the Q is diagonal.

[dkarrasch]

I realized my proposal is circular, so nonsense. But what would AbstractMatrix(::AbstractQ) do? Probably it shouldn't call Matrix(Q).

[dlfivefifty]

I think:

AbstractMatrix(Q::AbstractQ) = Matrix(Q)

should suffice. I can then overload for a specific subtype.

[dkarrasch]

Oh, you mean a specific subtype of AbstractQ, not of AbstractMatrix. Got it!

[dlfivefifty]

I do feel like the conflation of reduce QR and full QR is an inconsistent mess... maybe they should be separated via two different types...

It's disappointing that AbstractQ has no documentation at all. Does it represent square orthogonal operators? Or operators with orthogonal columns?

[dkarrasch]

It may not be great, but we have https://docs.julialang.org/en/v1/stdlib/LinearAlgebra/#man-linalg-abstractq. The default interpretation is as square orthogonal operators. For specific types, you need to overload Matrix{T}(::MyQType) to get potentially rectangular matrix representations. Also, one may overload qsize_check to allow for ambiguous sizes. One then gets the "append by zeros to the appropriate size" behavior for free. The fact that we append by zeros makes clear that at the end of the day, we use the Q's as square operators (as represented by Householder reflections), though some find that strange: #1172.