Suppose that firms in a particular industry fall into one of three size categories: large, medium and small. If a firm is large one year, the probabilities that it will remain large, fall into the medium size category, or become small in the next year are, respectively $0.7$, $0.2$, and $0.1$. For a firm of medium size, the corresponding probabilities are, respectively, $0.1$, $0.8$, and $0.1$. For a small firm, the probabilities are, respectively, $0$, $0.1$, and $0.9$.
I figured out the transition matrix stated below.
a) Express transition matrix $T = \begin{bmatrix} 0.7 & 0.1 & 0 \\ 0.2 & 0.8 & 0.1 \\ 0.1 & 0.1 & 0.9 \end{bmatrix}$ in the form $QDQ^{-1}$, where $D$ is a diagonal matrix.
b) Suppose that the total number of firms in the industry remains fixed at $4000$. By considering what happens to $D^n$ as $n \to \infty$, determine how many firms fall into each category in the long term.
(a) $D$ is the diagonal matrix of eigenvalues, and $Q$ is normalized eigenvectors. Perform the decomposition. If you do it right, $D$ will have $1, 0.8, 0.6$ on its diagonal.
(b) $D^n$ is the same diagonal matrix but the entrees will be $1^n = 1, 0.8^n \to 0, 0.6^n \to 0$ so the last remaining result will converge to the eigenvector corresponding to the eigenvalue 1.