P is the transition matrix of a regular Markov chain. Find the long range transition matrix L of P. $$ P = \begin{bmatrix} 1/2 & 1/4 & 1/4\\1/2&1/2 &1/4\\0 &1/4 & 1/2\end{bmatrix}$$
I found the eigenvalues to be $\lambda = 1$ and $\lambda = 1/4.$ Which means P is a transition matrix.
I am not sure how to go on with this question after this. Finding the eigenspace for $\lambda = 1$ gives me $ v = \begin{bmatrix} 1.5 \\ 2 \\ 1 \end{bmatrix}$ but I can't see how to turn that into a steady-state vector.