Consider the system x _ = Ax = 5 3 2 8 5 4 4 3 3 x. (39) a. Show that r = 1 is a triple

QUESTION:

Consider the system x _ = Ax = 5 3 2 8 5 4 4 3 3 x. (39) a. Show that r = 1 is a triple eigenvalue of the coefficient matrix A and that there are only two linearly independent eigenvectors, which we may take as (1) = 102 , (2) = 02 3 . (40) Write down two linearly independent solutions x(1) (t) and x(2) (t) of equation (39). b. To find a third solution, assume that x = tet + et ; then show that and must satisfy (A I) = 0, (41) (A I) = . (42) c. Equation (41) is satisfied if is an eigenvector, so one way to proceed is to choose to be a suitable linear combination of (1) and (2) so that equation (42) is solvable, and then to solve that equation for . However, let us proceed in a different way and follow the pattern of 15. First, show that satisfies (A I)2 = 0. Further, show that (AI)2 = 0. Thus can be chosen arbitrarily, except that it must be independent of (1) and (2) . d. A convenient choice for is = (0, 0, 1)T . Find the corresponding from equation (42). Verify that is an eigenvector of A. e. Write down a fundamental matrix (t) for the system (39). f. Form a matrix T with the eigenvector (1) in the first column and with the eigenvector from part d and the generalized eigenvector in the other two columns. Find T1 and form the product J = T1AT. The matrix J is the Jordan form of A. 1