I am currently trying to solve this question from a sample test for my linear algebra course:
After exposure to certain live pathogens, the body develops long-term immunity. The evolution over time of the associated disease can be modeled as a dynamical system whose state vector at time t consists of the number of people who have not been exposed and are therefore susceptible, the number who are currently sick with the disease, and the number who have recovered and are now immune. Suppose that the associated yearly 3 x 3 transition matrix has eigenvalues λ = 1, 1/2, 0, and that the eigenvectors corresponding to the first two eigenvalues are and x1 = (60, 20, 30) and x2 = (-60, -30, -90) respectively. The initial state vector for the population is given by v0 = 500*x1 + 200*x2 + 100*x3 where the third eigenvector x3 is not given here. How many people will be sick with the disease 2 years later?
I know that the probability vector at time t will be p(t) = T^t * p0 where T is the transition matrix and p0 is the initial probability vector.
Since I have been given two eigenvectors and three eigenvalues of the 3 x 3 transition matrix, I thought I could determine T^t by using diagonalisation: T^t = P* D^t * P^-1
However, the problem I have is not knowing the third eigenvector of T. This is what I think I have deduced so far:
T
\begin{bmatrix}60&-60&a\\20&-30&b\\30&-90&c\end{bmatrix} * \begin{bmatrix}1&0&0\\0&1/2&0\\0&0&0\end{bmatrix} * \begin{bmatrix}60&-60&a\\20&-30&b\\30&-90&c\end{bmatrix}^-1
How do I go about finding T without the third eigenvector using this approach? Is this the right approach for the question?
The third eigenvector does not matter, since the associated eigenvalue is $0$. You don’t need the explicit formula for T, it suffices to write that $$T x_1 = 1* x_1$$
$$T x_2 = \frac{1}{2} x_2$$
$$T x_3 = 0 * x_3$$
And by linearity of T,
$$T^t v_0 = 1^t *500* x_1 + \left(\frac{1}{2}\right)^t * 200* x_2 + 0 ^t*100* x_3 $$
Which gives the result $t$ years later.