Diagonalization of matrices

We live in a city with only 3 weather conditions: Foggy, Hot, and Stormy. Ordering the states as mentioned, the transition probabilities are: $A=\begin{bmatrix} 0.6 & 0.2 & 0.2 \\ 0.2 & 0.6 & 0.2 \\ 0.2 & 0.2 & 0.6 \\ \end{bmatrix}$
Matrix $A$ is diagonalizable with matrices $P = \begin{bmatrix} 1 & -1 & -1 \\ 1 & 1 & 0 \\ 1 & 0 & 1 \end{bmatrix}$, $D = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \frac{2}{5} & 0 \\ 0 & 0 & \frac{2}{5} \\ \end{bmatrix}$and $P^{-1} = \begin{bmatrix} \frac{1}{3} & \frac{1}{3} & \frac{1}{3} \\ -\frac{1}{3} & \frac{2}{3} & -\frac{1}{3} \\ -\frac{1}{3} & -\frac{1}{3} & \frac{2}{3} \end{bmatrix}$.

If it is hot today, what is the probability that the day after tomorrow there will be fog? Give the result to $2$ decimal places.