Issues with inv(Triangular)

[DanielVandH]

Consider

julia> using LazyArrays, ArrayLayouts, LinearAlgebra

julia> A = UpperTriangular(ApplyArray(inv, rand(5, 5)))
5×5 UpperTriangular{Float64, ApplyArray{Float64, 2, typeof(inv), Tuple{Matrix{Float64}}}}:
 2.35135   1.68449   -1.84848   -0.955078  -1.85021
  ⋅       -0.271777  -1.25844    0.94071   -1.27413
  ⋅         ⋅        -0.124992  -1.4524     2.35783
  ⋅         ⋅          ⋅         0.669905  -0.609163
  ⋅         ⋅          ⋅          ⋅         0.840739

julia> inv(A)
inv(5×5 UpperTriangular{Float64, ApplyArray{Float64, 2, typeof(inv), Tuple{Matrix{Float64}}}}):
  0.425287   2.63595  -32.8287   -74.2701    43.1852
 -0.0       -3.67949   37.0457    85.4846   -47.5317
 -0.0       -0.0       -8.00051  -17.3457     9.86934
  0.0        0.0        0.0        1.49275    1.08158
  0.0        0.0        0.0        0.0        1.18943

julia> MemoryLayout(inv(A))
LazyArrays.InvLayout{TriangularLayout{'U', 'N', LazyArrays.LazyLayout}}()

The LazyArrays.InvLayout{TriangularLayout{'U', 'N', LazyArrays.LazyLayout}}() could maybe be TriangularLayout{'L', 'N', LazyArrays.LazyLayout}()? (Actually doing

applylayout(::Type{typeof(inv)}, ::TriangularLayout{'U', UNIT, ML}) where {UNIT, ML} = TriangularLayout{'L', UNIT, ML}()
applylayout(::Type{typeof(inv)}, ::TriangularLayout{'L', UNIT, ML}) where {UNIT, ML} = TriangularLayout{'U', UNIT, ML}()

causes more issues with stackoverflows, so I guess the Inv should stay in the layout.) One issue that the current situation causes is e.g.

julia> using BandedMatrices

julia> colsupport(inv(A), 1)
Base.OneTo(5)

which is of course more problematic for infinite matrices.

[DanielVandH]

Actually this might be more on LazyArrays.jl than here since it comes from https://github.com/JuliaArrays/LazyArrays.jl/blob/8fc085dc65af7114b74c774da71a7942ffa2461f/src/linalg/inv.jl#L84