LU decomposition over RealDoubleField for non-square matrix is not LU

[unhyperbolic]

When doing LU decomposition over RealDoubleField with a non-square matrix, the L matrix has entries above the diagonal and non-1 on the diagonal and the U matrix has entries below it. Some entries are clearly too large...

sage: m=matrix([[0,0,0],[0,0,0],[1,0,0],[0,1,0],[0,0,1]])
sage: (m.change_ring(RDF) + RDF(0.000001) * matrix.random(RDF,5,3)).LU()
(
[0.0 0.0 1.0 0.0 0.0]  [                    1.0                     0.0                     0.0 -1.8760367664879336e-07                     1.0]  [      1.0000005169231803    5.204559874895986e-09   -4.501977663112735e-07]
[0.0 0.0 0.0 1.0 0.0]  [-1.8760367664879336e-07                     1.0                     0.0  -2.583294299022137e-07   5.728415115080068e-07]  [                     0.0       1.0000000471144364    9.669789257946877e-07]
[0.0 0.0 0.0 0.0 1.0]  [ -2.583294299022137e-07   5.728415115080068e-07                     1.0   4.766095912370668e-07 -3.5971730044271383e-07]  [                     0.0                      0.0       0.9999999362631513]
[0.0 1.0 0.0 0.0 0.0]  [  4.766095912370668e-07 -3.5971730044271383e-07  1.0203360856729682e-07  -2.910426049071506e-07  1.8384295593485092e-07]  [                     0.0        4.7439512333e-314        4.7705952805e-314]
[1.0 0.0 0.0 0.0 0.0], [ -2.910426049071506e-07  1.8384295593485092e-07  1.0357502158488482e-07       4.7362984647e-314                     0.0], [       8.7001827431e-313 -1.2907432083930526e-231  -3.105036184601418e+231]
)

Component: numerical

Issue created by migration from https://trac.sagemath.org/ticket/25820