Question

Question 16:Produce the general solution of the dynamical system \({x_{k + 1}} = A{x_k}\) when \(A\) is the stochastic matrix for the Hertz Rent A Car model in Exercise 16 of Section 4.9.

Short answer

The required general solution is \({\bf{x}}\left( t \right) = {c_1}{{\bf{v}}_1} + {c_2}{\left( {0.89} \right)^k}{{\bf{v}}_2} + {c_3}{\left( {0.81} \right)^k}{{\bf{v}}_3}\).

Step-by-step solution

Determine the eigenvalues and eigenvector of the matrix using MATLAB

From exercise 16, we have:

\(A = \left( {\begin{array}{*{20}{c}}{0.90}&{0.01}&{0.09}\\{0.01}&{0.90}&{0.01}\\{0.09}&{0.09}&{0.90}\end{array}} \right)\)

Enter this matrix in MATLAB as:

>> \(A = \left( {\begin{array}{*{20}{c}}{0.90\,\,0.01\,\,0.09;}&{0.01\,\,0.90\,\,0.01;}&{0.09\,\,0.09\,\,0.90}\end{array}} \right)\);

For eigenvalues, enter instruction as:

>> \(E = {\rm{eigs}}\left( A \right)\);

We get:

\(E = \left( {\begin{array}{*{20}{c}}{0.81}\\{0.89}\\{1.00}\end{array}} \right)\)

Now, for eigenvectors, enter instruction as:

>> \(\left( {\begin{array}{*{20}{c}}A&B\end{array}} \right) = {\rm{eigs}}\left( A \right)\);

So, we have:

\(\begin{array}{l}{v_1} = \left( {\begin{array}{*{20}{c}}{ - 0.6700}\\{ - 0.1399}\\{ - 0.7289}\end{array}} \right)\\{v_2} = \left( {\begin{array}{*{20}{c}}{0.7071}\\{ - 0.7071}\\{ - 0.0000}\end{array}} \right)\\{v_3} = \left( {\begin{array}{*{20}{c}}{ - 0.7071}\\{0.0000}\\{0.7071}\end{array}} \right)\end{array}\)

Thus, these are the required eigenvectors.

The General Solution to the system.

Now, the general solution can be given as:

\({\bf{x}}\left( t \right) = {c_1}{{\bf{v}}_1} + {c_2}{\left( {0.89} \right)^k}{{\bf{v}}_2} + {c_3}{\left( {0.81} \right)^k}{{\bf{v}}_3}\)

Hence, this is the required solution.