Question

Problem 4. (Discrete time dynamical system ). Consider the following discrete time dynamical system: Assume xo is given and 0.5 0.5 0.2 0.8 (a) Find eigenvalues of matrix A (b) For each eigenvalue find one eigenvector. (c) Let P be the matrix that has the eigenvectors as its columns. Find P-1 (d) Find P- AP (e) Use the answer from part (d) to find A" and xn-A"xo. (Your answers wl be in terms of n (f) Find xn and limn→ooXn for 0.3 0.7 0.9 0.1 and x0

Answer 1

1 Find eigemalues from the characteristic polynonmialo Details 2 For every λ we find its onn vector(s): ystem of linear equations, we soive it by Caussian Elimination Find the variablex from the equation 1 of the system(1) .d-A x E=0, So we havea homogeneous system. of linear equations, we solve it by Gaussian Elimination: 0C2 075 R2で):R-R' ( 0 Find the variablex from the equation 1 of the system(1) I-

$\\P = \begin{bmatrix} -2.5 &1 \\ 1& 1 \end{bmatrix},\ D = \begin{bmatrix} 0.3 &0 \\ 0 & 1 \end{bmatrix},\ P^{-1} = \begin{bmatrix} -0.29 & 0.29\\ 0.29 & 0.71 \end{bmatrix}\\$

$\\P = \begin{bmatrix} -2.5 &1 \\ 1& 1 \end{bmatrix},\ D = \begin{bmatrix} 0.3 &0 \\ 0 & 1 \end{bmatrix},\ P^{-1} = \begin{bmatrix} -0.29 & 0.29\\ 0.29 & 0.71 \end{bmatrix}\\ \\PAP^{-1}= D = \begin{bmatrix} 0.3 & 0\\ 0 &1 \end{bmatrix}\\$