For the transition matrix P= 0.7 0.3 0.3 0.7 , solve the equation SP = S to find the stationary matrix S and the limiting matrix P.

Step 1 ：Given the transition matrix P = \(\begin{bmatrix} 0.7 & 0.3 \\ 0.3 & 0.7 \end{bmatrix}\)

Step 2 ：We need to find the stationary matrix S that satisfies the equation SP = S. This is equivalent to solving the equation (SP - S) = 0, or equivalently, S(P - I) = 0, where I is the identity matrix.

Step 3 ：Solving the equation gives us the stationary matrix S = \(\begin{bmatrix} 0.5 \\ 0.5 \end{bmatrix}\)

Step 4 ：The limiting matrix P is the matrix that the transition matrix P converges to as the number of transitions goes to infinity. Since the transition matrix P is a regular Markov chain, the limiting matrix is simply the matrix with all rows equal to the stationary matrix S.

Step 5 ：Calculating the limiting matrix gives us P_limit = \(\begin{bmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{bmatrix}\)

Step 6 ：Final Answer: The stationary matrix S is \(\boxed{\begin{bmatrix} 0.5 & 0.5 \end{bmatrix}}\) and the limiting matrix P is \(\boxed{\begin{bmatrix} 0.5 & 0.5 \\ 0.5 & 0.5 \end{bmatrix}}\)