Suppose I have the followin situation :
The weather can either be "bad" or "good".
If the weather yesterday was "good" and the weather today is "good", then the probability that the weather will be "good" tomorrow is $p_1$
If the weather yesterday was "good" and the weather today is "bad", then the probability that the weather will be "good" tomorrow is $p_2$
If the weather yesterday was "bad" and the weather today is "good, then the probability that the weather will be "good" tomorrow is $p_3$
If the weather yesterday was "bad" and the weather today is "bad", then the probability that the weather will be "good" tomorrow is $p_4$
(The probability that the weather will be bad tomorrow is 1-$p_{...}$)
How can I represent it as a Markov chain and write a "useful" transition matrix (by useful, I mean such that I would be able to find the stationary vector (i.e. proportion of time spent in each "weather") ) ? I am looking for a general method. I did not found find much about it on the Web...
Thanks for help