Fix for #48911

1. Fix output permutation when A.P is not the identity matrix.
2. Fix the application of the elementary transformation (correct complex conjugation) to the solution as per LAPACK zlarzb https://netlib.org/lapack/explore-html/d5/ddd/zlarzb_8f_source.html
3. Align the final permutation with LAPACK's ZGELSY at https://netlib.org/lapack/explore-html/d8/d6e/zgelsy_8f_source.html
4. Fix a complex conjugate bug in the original solution for #48911, align with LAPACK zlarzb, https://netlib.org/lapack/explore-html/d5/ddd/zlarzb_8f_source.html
5. Improve permutation performance

stdlib/LinearAlgebra/src/qr.jl

@@ -601,19 +601,20 @@ function _wide_qr_ldiv!(A::QR{T}, B::AbstractMatrix{T}) where T
             B[m + 1:mB,1:nB] .= zero(T)
             for j = 1:nB
                 for k = 1:m
-                    vBj = B[k,j]
+                    vBj = B[k,j]'
                     for i = m + 1:n
-                        vBj += B[i,j]*R[k,i]'
+                        vBj += B[i,j]'*R[k,i]'
                     end
                     vBj *= τ[k]
-                    B[k,j] -= vBj
+                    B[k,j] -= vBj'
                     for i = m + 1:n
-                        B[i,j] -= R[k,i]*vBj
+                        B[i,j] -= R[k,i]'*vBj'
                     end
                 end
             end
         end
     end
+    B[A.p,:] = B[1:n,:]
     return B
 end