Markov chain with absorbing states?

Question

Let's say I have $5$ states (state $2$ to $6$, state $1$ is missing) when time$=0$, and $6$ states (state $1$ to $6$) when time$=1$, and now I want to calculate the transition matrix. Does it mean that the $[1,1]$ element in the matrix $p_{11}=1$, and $p_{12}$ to $p_{16} =0$? But in which case neither row- nor col-sums add up to one. Or can I create a transition matrix which is not square?

Link: https://stackoverflow.com/questions/34050222/r-add-missing-rows-columns-not-using-for-loop?noredirect=1#comment55855104_34050222

Answer 1

Let your $n\times n$ transition matrix be denoted $M(t)$ (as a function of time). (i) For all $t\geq0$, $M(t)$ is a right-stochastic matrix and (ii) for $t<1$, $M(t)$ must be of the form $$ M(t)=\left[\begin{array}{cc} 1 & 0\\ 0 & M^{\prime}(t) \end{array}\right]\text{ where }M^{\prime}(t)\text{ is an }(n-1)\times(n-1)\text{ right-stochastic matrix}. $$