Let $T$ be a $N\times N$ transition matrix for a markov chain with $N$ states. Thus $T_{ij}$ is the probability of transition from state $i$ to state $j$ (and thus rows summing to one). Now consider the matrix $$T_k=T\otimes T$$. Its easy to see that rows of $T_k$ sum to one and each entry is nonnegative. Thus, $T_k$ is a transition matrix for some markov chain which has $N^2$ states. What is the relation between markov chains corresponding to $T$ and $T_k$. I am not able to visualize this.
I think you just have a pair of independent chains. The probability of making a transition from (i,j) to (k,l) is $p_{ik}p_{jl}$, where the first component is the state of the first chain and the second of the second. The transitions out of the (i,j) state are found in the N(i1) + j th row.

$\begingroup$ can you make any comment on the relation between the steady state probability vectors of both the chains? $\endgroup$ – dineshdileep Dec 1 '16 at 6:26

$\begingroup$ Regarding the comment above, if $\pi$ is the stationary/invariant distribution, then $\pi \otimes \pi$ is the stationary distribution for the product chain. This is because $$ (\pi \otimes \pi)(T \otimes T) = (\pi T) \otimes (\pi T) = \pi \otimes \pi. $$ The first equality is just the "mixedproduct" property of Kronecker products. $\endgroup$ – Taylor Feb 21 '19 at 3:28