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rmatrixeigenvalueeigenvector

eigen() and the correct eigenvectors

My problem is the following:

I'm trying to use R in order to compute numerically this problem. So I've correctly setup the problem in my console, and then I tried to compute the eigenvectors.

R program

But I expect that the eigenvector associated with lambda = 1 is (1,2,1) instead of what I've got here. So, the scaling is correct (0.4082483 is effectively half of 0.8164966), but I would like to obtain a consistent result.

My original problem is to find a stationary distribution for a Markov Chain using R instead of doing it on paper. So from a probabilistic point of view, my stationary distribution is a vector whose sum of the components is equal to 1. For that reason I was trying to change the scale in order to obtain what I've defined "a consistent result".

How can I do that ?

Solution

  • The eigen vectors returned by R are normalized (for the square-norm). If V is a eigen vector then s * V is a eigen vector as well for any non-zero scalar s. If you want the stationary distribution as in your link, divide by the sum:

    V / sum(V)

    and you will get (1/4, 1/2, 1/4).

    So:

    ev <- eigen(t(C))$vectors
ev / colSums(ev)

    to get all the solutions in one shot.