CODEWITHSUNDEEP
pythonmatlabnumpylinear-algebrasympy
carobnodrvo
carobnodrvo

Reputation: 1051

SymPy rank different from NumPy matrix rank

Given some SymPy matrix M

M = Matrix([
 [0.000111334436666596, 0.00114870370895408, -0.000328330524152990, 5.61388353859808e-6, -0.000464532588930332, -0.000969955779635878, 1.70579589853818e-5, -5.77891177019884e-6, -0.000186812539472235, -2.37115911398055e-5],
 [-0.00105346453420510, 0.000165063406707273, -0.00184449574409890, 0.000658080565333929, 0.00197652092300241, 0.000516180213512589, 9.53823860082390e-5, 0.000189858427211978, -3.80494288487685e-5, 0.000188984043643408],
 [-0.00102465075104153, -0.000402915220398109, 0.00123785300884241, -0.00125808154543978, 0.000126618511490838, 0.00185985865307693, 0.000123626008509804, 0.000211557638637554, 0.000407232404255796, 1.89851719447102e-5],
 [0.230813497584639, -0.209574389008468, 0.742275067362657, -0.202368828927654, -0.236683258718819, 0.183258819107153, 0.180335891933511, -0.530606389541138, -0.379368598768419, 0.334800403899511],
 [-0.00102465075104153, -0.000402915220398109, 0.00123785300884241, -0.00125808154543978, 0.000126618511490838, 0.00185985865307693, 0.000123626008509804, 0.000211557638637554, 0.000407232404255796, 1.89851719447102e-5],
 [0.00105346453420510, -0.000165063406707273, 0.00184449574409890, -0.000658080565333929, -0.00197652092300241, -0.000516180213512589, -9.53823860082390e-5, -0.000189858427211978, 3.80494288487685e-5, -0.000188984043643408],
 [0.945967255845168, -0.0468645728473480, 0.165423896937049, -0.893045423193559, -0.519428986944650, -0.0463256408085840, -0.0257001217930424, 0.0757328764368606, 0.0541336731317414, -0.0477734271777646],
 [-0.0273371493900004, -0.954100482348723, -0.0879282784854250, 0.100704543595514, -0.243312734473589, -0.0217088779350294, 0.900584332231093, 0.616061129532614, 0.0651163853434486, -0.0396603397583054],
 [0.0967584768347089, -0.0877680087304911, -0.667679934757176, -0.0848411039101494, -0.0224646387789634, -0.194501966574153, 0.0755161040544943, 0.699388977592066, 0.394125039254254, -0.342798611994521],
 [-0.000222668873333193, -0.00229740741790816, 0.000656661048305981, -1.12277670771962e-5, 0.000929065177860663, 0.00193991155927176, -3.41159179707635e-5, 1.15578235403977e-5, 0.000373625078944470, 4.74231822796110e-5]
 ])

I have calculated SymPy rank() and rref() of the matrix. Rank is 7 and rref() result is:

Matrix([
[1, 0, 0, 0, 0, 0, 0, -5.14556976678473, -3.72094268951566,  3.48581267477014],
[0, 1, 0, 0, 0, 0, 0, -5.52930150663022, -4.02230308325653,  3.79193678096199],
[0, 0, 1, 0, 0, 0, 0,  2.44893308665325,  1.83777402439421, -1.87489784909824],
[0, 0, 0, 1, 0, 0, 0, -7.33732284392352, -5.25036238623229,  4.97256759287563],
[0, 0, 0, 0, 1, 0, 0,  5.48049237370489,  3.90091366576548, -3.83642187384021],
[0, 0, 0, 0, 0, 1, 0, -10.6826798792866, -7.56560803870182,  7.45974067056387],
[0, 0, 0, 0, 0, 0, 1, -3.04726210012149, -2.66388837034592,  2.48327234504403],
[0, 0, 0, 0, 0, 0, 0,                 0,                 0,                 0],
[0, 0, 0, 0, 0, 0, 0,                 0,                 0,                 0],
[0, 0, 0, 0, 0, 0, 0,                 0,                 0,                 0]])

Weird thing is that if I calculate rank with either NumPy or MATLAB I get value 6 and calculating rref with MATLAB I get the expected result - last 4 rows are all zero (instead of only last 3).

Does any one know where does this difference comes from and why am I unable to get correct results with SymPy? I know that rank 6 is correct because it is system of the equations where some linear dependency exist.

Upvotes: 5

Views: 2636

Answers (1)

Matthieu Brucher
Matthieu Brucher

Reputation: 22023

Looking at the eigenvalues of your matrix, the rank is indeed 6:

array([ 1.14550481e+00+0.00000000e+00j, -1.82137718e-01+6.83443168e-01j,
   -1.82137718e-01-6.83443168e-01j,  2.76223053e-03+0.00000000e+00j,
   -3.51138883e-04+8.61508469e-04j, -3.51138883e-04-8.61508469e-04j,
    5.21160131e-17+0.00000000e+00j, -2.65160469e-16+0.00000000e+00j,
   -2.67753616e-18+9.70937977e-18j, -2.67753616e-18-9.70937977e-18j])

With the sympy version I have, I get even a rank of 8, compared to the rank 6 that numpy returns.

But actually, Sympy cannot solve the eigenvalues of this matrix due to the size of the matrix (probably related to SymPy could not compute the eigenvalues of this matrix).

So one of them, Sympy, is trying to solve symbolically the equations and find the rank (based on imperfect floating point numbers), whereas the other one, numpy, uses approximations (lapack IIRC) to find the eigenvalues. By having an adequate threshold, numpy finds the proper rank, but it could have said differently with a different threshold. Sympy tried to find the rank based on an approximate system of a perfect 6 rank system and finds that it is of rank 7 or 8. It's not surprising due to the floating point difference (Sympy moves to integers to try to find the eigenvalues, for instance, instead of staying in floating point realm).

Upvotes: 4

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