Find the eigenvalues of A. Then determine the possible Jordan canonical forms of A (up to permutation of the blocks). ii) Compute the Jordan canonical form J of A. Give a transformation matrix, namely an invertible matrix S such that S?1AS = J, and compute explicitely its inverse S?1. iii) Recall that the solution to the initial value problem dx = Ax, x(0) = x0, dt with x0 ? R3 a given intial condition, is x(t) = etA x0 . First compute etJ . Then determine all possible initial states x0 for which ?x(t)?2 remains bounded as t ? ?. For such states, what is the limit of x(t) as t ? 0?

Let 2 A = 3 i) Find the eigenvalues of A. Then determine the possible Jordan canonical forms of A (up to permutation of the blocks). ii) Compute the Jordan canonical form J of A. Give a transformation matrix, namely an invertible matrix S such that S-AS = J, and compute explicitely its inverse S-1. iii) Recall that the solution to the initial value problem dx dt Ax, x(0) = XO; with xo ( R a given intial condition, is x(t) = etAxo. First compute et. Then determine all possible initial states Xo for which (x(t) |2 remains bounded as t -+ co. For such states, what is the limit of x(t) as t - 0