Eigendecomposition determination error
I have a very simple question. This is due to a computational tolerance error.
Let me make (see the end) my own representation of matrix A in eigenvector V and diagonal eigenvalues of D and construct it again by multiplying V ^ -1 * D * V.
The obtained value is far from A, the error is quite large.
I would like to know if I am using the wrong functions to accomplish this task, or at least how to mitigate this error. Thank you in advance
in[1]:import numpy
from scipy import linalg
A=matrix([[16,-9,0],[-9,20,-11],[0,-11,11]])
D,V=linalg.eig(A)
D=diagflat(D)
matrix(linalg.inv(V))*matrix(D)*matrix(V)
out[1]:matrix([[ 15.52275377, 9.37603361, 0.79257097],
[9.37603361, 21.12538282, -10.23535271],
[0.79257097, -10.23535271, 10.35186341]])
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Isn't that so? A*V = V*D
from the definition, therefore A = V*D*V^(-1)
.
>>> import numpy as np
>>> from scipy import linalg
>>> A = np.matrix([[16,-9,0],[-9,20,-11],[0,-11,11]])
>>> D, V = linalg.eig(A)
>>> D = np.diagflat(D)
>>>
>>> b = np.matrix(linalg.inv(V))*np.matrix(D)*np.matrix(V)
>>> b
matrix([[ 15.52275377+0.j, 9.37603361+0.j, 0.79257097+0.j],
[ 9.37603361+0.j, 21.12538282+0.j, -10.23535271+0.j],
[ 0.79257097+0.j, -10.23535271+0.j, 10.35186341+0.j]])
>>> np.allclose(A, b)
False
but
>>> f = np.matrix(V)*np.matrix(D)*np.matrix(linalg.inv(V))
>>> f
matrix([[ 1.60000000e+01+0.j, -9.00000000e+00+0.j, -9.54791801e-15+0.j],
[ -9.00000000e+00+0.j, 2.00000000e+01+0.j, -1.10000000e+01+0.j],
[ -1.55431223e-15+0.j, -1.10000000e+01+0.j, 1.10000000e+01+0.j]])
>>> np.allclose(A, f)
True
Also, there are recipes to use np.dot
to avoid all these matrix conversions like
>>> dotm = lambda *args: reduce(np.dot, args)
>>> dotm(V, D, inv(V))
array([[ 1.60000000e+01+0.j, -9.00000000e+00+0.j, -9.54791801e-15+0.j],
[ -9.00000000e+00+0.j, 2.00000000e+01+0.j, -1.10000000e+01+0.j],
[ -1.55431223e-15+0.j, -1.10000000e+01+0.j, 1.10000000e+01+0.j]])
which I often find cleaner, but YMMV.
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