[SciPy-User] Eigenvalue decomposition bug

[Xavier Saint-Mleux]

Sebastian Walter wrote:
> Hello Pauli,
> On what kind of matrix did you observe such unstable behavior?
> Were there repeated eigenvalues?
>   

It happens to me a lot with complex covariance matrices (Hermitian). 
Here's a simple example that returns non-real eigenvalues for an
Hermitian matrix:


>>> np.random.seed(0)
>>> x = np.random.random((3,3)) + np.random.random((3,3)) * 1j
>>> x = (x+x.T.conj())/2 # make it Hermitian
>>> x == x.T.conj() # ensure it is Hermitian
array([[ True,  True,  True],
       [ True,  True,  True],
       [ True,  True,  True]], dtype=bool)
>>> np.linalg.eigvals(x) # returns complex values
array([ 1.99062044 -4.98523579e-17j,  0.18062978 -9.36952928e-19j,
       -0.23511915 -2.19606549e-17j])
>>> np.linalg.eigvalsh(x) # imag always zero
array([-0.23511915+0.j,  0.18062978+0.j,  1.99062044+0.j])
>>>



Xavier



> Sebastian
>
>
>
>
>
> On Tue, May 11, 2010 at 10:39 PM, Pauli Virtanen <pav at iki.fi> wrote:
>   
>> Tue, 11 May 2010 16:04:00 -0400, Ian Goodfellow wrote:
>>     
>>> I've find that (scipy/numpy).linalg.eig have a problem where given a
>>> symmetric matrix they return complex eigenvalues. I can use scipy.io to
>>> save this matrix in matlab format, load it in matlab, and use matlab's
>>> eig function to succesfully decompose it with real eigenvalues, so the
>>> problem seems to be with scipy/numpy or their dependencies, not with my
>>> matrix. Is this a known issue? And is there a good workaround?
>>>       
>> Use the eigh function if you know your matrix is symmetric.
>>
>> Matlab IIRC checks first if the matrix is symmetric, and if yes, uses a
>> symmetric-specific eigensolver. Numpy and Scipy don't do this automatic
>> check.
>>
>> A nonsymmetric eigensolver cannot know that your matrix is supposed to
>> have real eigenvalues, so it's possible some of them explode to complex
>> pairs because of minuscule numerical error. The imaginary part, however,
>> is typically small.
>>
>> --
>> Pauli Virtanen
>>
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>>     
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