1.6. Exercises 25 is an example of a phase portrait of a differential equation. Observe that by uniqueness of solutions, these parametrized curves cannot intersect. De- scribe the solutions as functions of time t R and the phase portraits in the plane R2 of ˙ = Ax for A = −1 0 0 −3 , A = 1 0 0 −1 , A = 3 0 0 1 . What are the relations between the corresponding solutions? Exercise 1.6.2. Describe the solutions as functions of time t R and the phase portraits in the plane R2 of ˙ = Ax for A = 1 1 0 1 , A = 0 1 0 0 , A = −1 1 0 −1 . Exercise 1.6.3. Describe the solutions as functions of time t R and the phase portraits in the plane R2 of ˙ = Ax for A = 1 −1 1 1 , A = 0 −1 1 0 , A = −1 −1 1 −1 . Exercise 1.6.4. Describe all possible phase portraits in R2 taking into account the possible Jordan blocks. Exercise 1.6.5. Compute the solutions of ˙ = Ax for A = 1 1 0 0 1 0 0 0 2 . Exercise 1.6.6. Determine the stable, center, and unstable subspaces of ˙ = Ax for A = 3 −4 1 1 0 −1 −1 4 −3 . Exercise 1.6.7. Show that for a matrix A gl(d, R) and T 0 the spec- trum spec(eAT ) is given by {eλT | λ spec(A)}. Show also that the maximal dimension of a Jordan block for μ spec(eAT ) is given by the maximal di- mension of a Jordan block of an eigenvalue λ spec(A) with eλT = μ. Take into account that eıνT = eıν T for real ν, ν does not imply ν = ν . As an example, discuss the eigenspaces of A and the eigenspace for the eigenvalue 1 of eAT with A given by A = 0 0 0 0 0 −1 0 1 0 .
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