4.F.3 Suppose the eigenvalues of a 2 x 2 matrix A are 3 and 5, with corresponding eigenvectors 3 - - [] --[:) = Vj = U2

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4 F 3 Suppose The Eigenvalues Of A 2 X 2 Matrix A Are 3 And 5 With Corresponding Eigenvectors 3 Vj U2 1 (10.84 KiB) Viewed 37 times

4 F 3 Suppose The Eigenvalues Of A 2 X 2 Matrix A Are 3 And 5 With Corresponding Eigenvectors 3 Vj U2 2 (58.73 KiB) Viewed 37 times

4.F.3 Suppose the eigenvalues of a 2 x 2 matrix A are 3 and 5, with corresponding eigenvectors 3 - - [] --[:) = Vj = U2 =

a) Find the matrix A. Hint: A = PDP-1 Note: you don't need to know A for the rest of this problem. b) Sketch the trajectories of this dynamical system, 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). c) Classify the origin as a stable or unstable equilibrium. Classify the system as a node or saddle. Alternatively, if you prefer, you can use the book's nomenclature: classify the origin as an attractor (stable node), repeller (unstable node), or saddle point. d) Pick a random point (not on a special trajectory) to be co, and describe the behavior of <k as ko, and ask +-0. (See parenthetical in 4.F.1.)