# How do I get the average transition matrix for three consecutive years?

I have a one year transition matrix for three consecutive years. Multiplying these three matrices together yields the three year transition matrix. I want to obtain the average transition matrix for the three years (average^3 = 3yrtransition)

What is the procedure to be used? Is this possible at all? (I kind of realize that there might be multiple solutions to this problem due to possible multiple paths to achieve the end state).

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## 1 Answer

If the transition matrix has distinct eigenvalues, you can diagonalize it and then take the cube root of the diagonal. E.g., you can compute the SVD, verify that the eigenvalues are distinct, take the cube root of the diagonal matrix, then re-multiply it together.

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Johann, is your procedure possible if the transition matrix has complex eigenvalues? – morsecode May 20 '11 at 0:02
For most, yes, it should. It will work for any normal matrix. – Johann Hibschman May 23 '11 at 19:42