Let 1 -2 1 A = 0 0 -1 -1 -1 i) Find the eigenvalues of...

 

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Let 1 -2 1 A = 0 0 -1 -1 -1 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 = Ax, dt x(0) = x0, 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 . For such states, what is the limit of x(t) as t 0? Let 1 -2 1 A = 0 0 -1 -1 -1 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 = Ax, dt x(0) = x0, 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 . For such states, what is the limit of x(t) as t 0?