Consider the system x = Ax = 5 3 2 8 5 4 433 x. (i) (a) Show that r = 1 is a triple

Chapter 7, Problem 18

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Consider the system x = Ax = 5 3 2 8 5 4 433 x. (i) (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) = 1 0 2 , (2) = 0 2 3 . (ii) Find two linearly independent solutions x(1) (t) and x(2) (t) of Eq. (i). (b) To find a third solution, assume that x = tet + et ; then show that and must satisfy (A I) = 0, (iii) (A I) = . (iv) (c) Show that = c1 (1) + c2 (2) , where c1 and c2 are arbitrary constants, is the most general solution of Eq. (iii). Show that in order for us to solve Eq. (iv), it is necessary that c1 = c2.(d) It is convenient to choose c1 = c2 = 2. For this choice, show that =242 , =001 , (v)where we have dropped the multiples of (1) and (2) that appear in . Use the resultsgiven in Eqs. (v) to find a third linearly independent solution x(3)(t) of Eq. (i).(e) Write down a fundamental matrix (t) for the system (i).(f) Form a matrix T with the eigenvector (1) in the first column and with the eigenvector and the generalized eigenvector from Eqs. (v) in the other two columns. Find T1 andform the product J = T1AT. The matrix J is the Jordan form of A.