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I'm trying to calculate the eigenvectors and eigenvalues of this matrix
import numpy as np
la = 0.02
mi = 0.08
n = 500
d1 = np.full(n, -(la+mi), np.double)
d1[0] = -la
d1[-1] = -mi
d2 = np.full(n-1, la, np.double)
d3 = np.full(n-1, mi, np.double)
A = np.diagflat(d1) + np.diagflat(d2, -1) + np.diag(d3, 1)
e_values, e_vectors = np.linalg.eig(A)
If I set the dimensions of the matrix to n < 110 the output is fine. However, if I set it to n >= 110 both the eigenvalues and the eigenvector components become complex numbers with significant imaginary parts. Why does this happen? Is it supposed to happen? It is very strange behavior and frankly I'm kind of stuck.
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For instance, every complex matrix has an eigenvalue. Every real matrix has an eigenvalue, but it may be complex. In fact, a field K is algebraically closed iff every matrix with entries in K has an eigenvalue. You can use the companion matrix to prove one direction.
This is very easy to see; recall that if an eigenvalue is complex, its eigenvectors will in general be vectors with complex entries (that is, vectors in Cn, not Rn). If λ ∈ C is a complex eigenvalue of A, with a non-zero eigenvector v ∈ Cn, by definition this means: Av = λv, v = 0.
As a consequence of the fundamental theorem of algebra as applied to the characteristic polynomial, we see that: Every n × n matrix has exactly n complex eigenvalues, counted with multiplicity. We can compute a corresponding (complex) eigenvector in exactly the same way as before: by row reducing the matrix A − λ I n .
If two matrices have the same set of eigenvectors but different eigenvalues, then they can be simultaneously diagonalized, which means that the two matrices commute which each other, that is if the two matrices are A and B, AB = BA.
What you are seeing appears to be fairly normal roundoff error. This is an unfortunate result of storing floating point numbers with a finite precision. It naturally gets relatively worse for large problems. Here is a plot of the real vs. imaginary components of the eigenvalues:
You can see that the imaginary numbers are effectively noise. This is not to say that they are not important. Here is a plot of the imaginary vs. real part, showing that the ratio can get as large as 0.06 in the worst case:
This ratio changes with respect to the absolute and relative quantities la and mi. If you multiply both by 10, you get
If you keep la = 0.02 and set mi = 0.8, you get a smaller imaginary part:
Things get really weird when you do the opposite, and increase la by a factor of 10, keeping mi as-is:
The relative precision of the calculation decreases for smaller eigenvalues, so this is not too surprising.
Given the relatively small magnitudes of the imaginary parts (at least for the important eigenvalues), you can either take the magnitude or the real part of the result since you know that all the eigenvalues are real.
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