4.W.1 Let A be the matrix below, and consider the dynamical system Ik+1 = Atk. 2 A 10 1=[ 1 ] 3 For this problem you wan

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4 W 1 Let A Be The Matrix Below And Consider The Dynamical System Ik 1 Atk 2 A 10 1 1 3 For This Problem You Wan 1 (99.52 KiB) Viewed 21 times

4.W.1 Let A be the matrix below, and consider the dynamical system Ik+1 = Atk. 2 A 10 1=[ 1 ] 3 For this problem you want graph paper which ranges at least from –85158 and -8 5458. a) Find the eigenvalues of A. Show that the origin is a saddle point for this dynamical system. b) Find the eigenspaces of A and graph these lines on a plane. = = [ ].compute 1 , 22, and 2-1. Show your work. : c) Consider the trajectory with 20 21. For the remaining computations in this problem, you can use computer algebra software (e.g. google for symbolab). Don't spend your time doing too much matrix multiplication. d) Carefully plot xk for -2 5k < 2 on your graph. Then sketch the trajectory followed by Ik for all integers k. It will help to compute 23 and 2-3, even if you don't sketch them, to see where things go next. , an e) Now consider the trajectory with 20 = and plot xx for -2 Sk < 1. Sketch the trajectory. Again, it will help to compute x2 and x-3 to see where things go next. f) Loosely sketch other trajectories, including at least one trajectory with each kind of behavior. Include all special trajectories (i.e. trajectories of eigenvectors). Trajectories should have arrows indicating direction (as k increases).