Eigenvalues and Eigenvectors

If we now have , which represents the amounts of students on a certain day, then represents the amounts of students on the next day.

After calculation we figure out that 0, 0.3, 0.6 and 1 are the Eigenvalues of A. With the help on the process page we now know that A is similar to D, which is represented below.

$$\mathbf D = \left( \begin{array}{cccc}0 & 0 & 0 & 0\\ 0 & 0.3 & 0& 0\\ 0 & 0 & 0.6 & 0\\ 0 & 0 & 0 & 1 \end{array} \right)$$

The next step is to calculate the Eigenvectors. They are shown below.

$$\vec b_1 = \left( \begin{array}{c}3\\ -18\\ 17\\ -2 \end{ar... ...4 = \left( \begin{array}{c}11\\ 4\\ 9\\ 16\end{array}\right)$$

Thus B is the following :

$$\mathbf B = \left( \begin{array}{cccc}3 & 0 & 3 & 11\\ -18& 0 & 0& 4\\ 17& 1 & 4& 9\\ -2& -1 & 1 & 16 \end{array} \right)$$