Left eigenvectors not giving correct (markov) stationary probability in scipy

Question

Given the following Markov Matrix: ``` import numpy, scipy.linalg A = numpy.array([[0.9, 0.1],[0.15, 0.85]]) ``` The stationary probability exists and is equal to `[.6, .4]`. This is easy to verify by taking a large power of the matrix: ``` B = A.copy() for _ in xrange(10): B = numpy.dot(B,B) ``` Here `B[0] = [0.6, 0.4]`. So far, so good. According to wikipedia: A stationary probability vector is defined as a vector that does not change under application of the transition matrix; that is, it is defined as a left eigenvector of the probability matrix, associated with eigenvalue 1: So I should be able to calculate the left eigenvector of `A` with eigenvalue of 1, and this should also give me the stationary probability. Scipy's implementation of `eig` has a left keyword: ``` scipy.linalg.eig(A,left=True,right=False) ``` Gives: ``` (array([ 1.00+0.j, 0.75+0.j]), array([[ 0.83205029, -0.70710678], [ 0.5547002 , 0.70710678]])) ``` Which says that the dominant left eigenvector is: `[0.832050…

Accepted Answer

The `[0.83205029, 0.5547002]` is just `[0.6, 0.4]` multiplied by ~1.39.

Although from "physical" point of view you need eigenvector with sum of its components equal 1, scaling eigenvector by some factor does not change it's "eigenness":

If , then obviously

So, to get `[0.6, 0.4]` you should do:

```
>>> v = scipy.linalg.eig(A,left=True,right=False)[1][:,0]
>>> v
array([ 0.83205029,  0.5547002 ])
>>> v / sum(v)
array([ 0.6,  0.4])
```

Left eigenvectors not giving correct (markov) stationary probability in scipy

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