Comment #2 on issue 2193 by [email protected]: Matrix.eigenvects() return
empty vectors in some cases
http://code.google.com/p/sympy/issues/detail?id=2193
(for information only)
I have investigated this issue and here my preliminary estimates
that it is because of sequentially related problems:
1. Solving of characteristic polynomials return not simplified roots
>>> roots(x**2 - (1 + I)*x + I, x)
{1/2 + I/2 - (-1)**(3/4)*2**(1/2)/2: 1, 1/2 + I/2 +
(-1)**(3/4)*2**(1/2)/2: 1}
>>> roots(x**2 - (1 + a)*x + a, x)
{1/2 + a/2 - (1 - 2*a + a**2)**(1/2)/2: 1, 1/2 + a/2 + (1 - 2*a +
a**2)**(1/2)/2: 1}
but expected {1: 1, I: 1} and {1: 1, a: 1} respective.
so Matrix.nullspace() can't operate with them (to determine zeros)
properly: substitution them to some expressions do not result zero.
1.1. In case of complex numbers it is because of:
>>> sqrt(-2*I)
(-1)**(3/4)*2**(1/2)
but expected (S.One - I) or (- S.One + I).
Also substitution of the one of the roots in polynomial raise an
Exception RuntimeError: 'maximum recursion depth...'
>>> p = x**2 - (1 + I)*x + I
>>> solution = roots(p, x)
>>> x1 = solution.iteritems().next()[0]
>>> x1
1/2 + I/2 - (-1)**(3/4)*2**(1/2)/2
# expected I or 1
>>> (x1**x1).expand()
Exception RuntimeError: 'maximum recursion ...'
# expected zero
1.2. In case of symbolic algebra, this is because of insufficient work
with roots_quadratic in sympy.polys.polyroots calculation determinant in
this case
>>> sqrt(1 + 2*a + a**2)
(1 + 2*a + a**2)**(1/2)
My conclusions:
I have seen through sympy-issues that work with polynomial is intensively
continuing now (and many issues NeedsReview), what about complex numbers
many issues age hanged.
I am not going to consider this issue to be closed right now - it must be
examined more, as well as situation with complex numbers (probably to solve
them at first).
But now I insert a necessary line of code for raising NotImplementedError
when eigen vectors can't be obtained (for any unrecognized reasons) to
catch this at the right time (pull request 115).
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