Exercise 7

Above we define a transition matrix to have each entry nonnegative and each column sum to $1$.

• Check that the three transition matrices shown in this Topic meet these two conditions. Must any transition matrix do so?

• Observe that if $A\vec{v}_{0}=\vec{v}_{1}$ and $A\vec{v}_{1}=\vec{v}_{2}$ then $A^{2}$ is a transition matrix from $\vec{v}_{0}$ to $\vec{v}_{2}$. Show that a power of a transition matrix is also a transition matrix.

• Generalize the prior item by proving that the product of two appropriately-sized transition matrices is a transition matrix.