- #1
Adel Makram
- 635
- 15
Homework Statement
Given an initial distribution state vector that represents the probability of the system to be in one of its states. Also given a Markov transition matrix. How to calculate the state vector of the system after n-transition?
Homework Equations
Assuming the initial state vector is a column vector x0, in literature it is considered as a row vector x0T. The state vector after the first transition will be: $$P x_0=x_1$$
where P is the Markov transitional matrix. After n-transitions, $$P^n x_0=x_n$$
The Attempt at a Solution
I tried to decompose P for an easier calculation using singular value decomposition, assuming that it is asymmetric matrix. $$P=UΣV^T$$.
$$P^n=(UΣV^T)^n=(U)^n (Σ)^n (V^T)^n$$.
I know that Σn is a diagonal matrix. Also I know that Un=I. But, I do not know the behavior of (VT)n.
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