I have a symbolic matrix at the start and I use the determinant to find roots, ok. But when I input some solution of the determinant equation in the matrix again, sage is not able to find the kernel (null space). Here's my atempt:
A=matrix([[x^2,-1.75*x,0],[-1.75*x,x^2,2.07*i*x^2+4.65*i],[0,2.07*i*x^2+-4.65*i,x^2]])
sol=solve(A.det(),x,solution_dict=True)
A.subs(x=sol[0][x]).right_kernel_matrix()
My output is simple:
[]
If you are mixing exact values with unexact ones it is unlikely that you get your answer. You have two solutions:
1.75, 2.07, etc by rational values 7/4, 207/100To illustrate the first option using scipy (from the method at http://scipy-cookbook.readthedocs.io/...)
sage: As = A.subs(x=sol[0][x]).change_ring(CDF)
sage: As
[ 2.333101328217149 2.673036254461398 0.0]
[ 2.673036254461398 2.333101328217149 9.479519749409498*I]
[ 0.0 0.179519749409498*I 2.333101328217149]
sage: As.det() # small but not zero
-3.1648558730042146e-14
sage: import scipy.linalg as linalg
sage: u, s, vh = linalg.svd(As.numpy()) # computing singular values decomposition
sage: s
array([ 1.04215568e+01, 3.45036344e+00, 8.81146174e-16])
From above you see that the small singular value is the last one (that is correspond to the "approximate kernel"). Hence to get the corresponding vector into Sage you can just do
sage: k = vector(vh[2])
sage: k
(-0.7524244881834203, 0.6567372851130923, -0.050532444272992305*I)
And you can check that it is in the kernel
sage: As * k
(6.661338147750939e-16, 1.27675647831893e-15, -1.0547118733938987e-15*I)
Asked: 2018-03-09 21:33:07 +0100
Seen: 452 times
Last updated: Mar 10 '18
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I believe that the function
solveis wrong by giving you exact symbolic expressions for the solutionsAnd this is part of the confusion here (it make no sense to substitute these "exact solutions" in your floating point matrix). I opened the trac ticket #24939 for this issue.