Obtaining a projection onto the range of a matrix through its QR decomposition

abulak · July 22, 2024, 12:25pm

this is what I did with svd and qr:

kalmarek/SymbolicWedderburn.jl/blob/master/src/image_basis.jl#L82-L103


      
          function image_basis!(
              A::AbstractMatrix{T},
              ::Nothing,
          ) where {T<:Union{AbstractFloat,Complex}}
              F = svd!(A)
              tol = _eps(T) * first(F.S)
              rk = count(x -> x > tol, F.S)
              return Matrix((@view F.U[1:rk, :])')
          end
          
          function image_basis!(
              A::AbstractSparseMatrix{T},
              ::Nothing,
          ) where {T<:Union{AbstractFloat,Complex}}
              F = qr(A)
              rk = rank(F)
              img = let tmp = F.Q * Matrix(I, size(A, 1), rk)
                  pinv = getfield(F, :rpivinv)
                  sparse((@view tmp[pinv, :])')
              end

This file has been truncated. show original

Incidentally I do know the mathematical rank beforehand and for my examples this numerical rank(F) has always agreed.

Matrices I deal with are of sizes up to 60k (of course sparse ;).

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Obtaining a projection onto the range of a matrix through its QR decomposition

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