Consider the systemx= Ax =5 3 28 5 4433 x. (i)(a) Show

Chapter 7, Problem 19

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Consider the systemx= Ax =5 3 28 5 4433 x. (i)(a) Show that r = 1 is a triple eigenvalue of the coefficient matrix A and that there areonly two linearly independent eigenvectors, which we may take as(1) =102 , (2) =023 . (ii)Write down 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) Equation (iii) 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 Eq. (iv) is solvable, and thento solve that equation for . However, let us proceed in a different way and follow thepattern of 17. First, show that satisfies(A I)2 = 0.Further, show that (A I)2 = 0. Thus can be chosen arbitrarily, except that it must beindependent of (1) and (2).(d) A convenient choice for is = (0, 0, 1)T . Find the corresponding from Eq. (iv).Verify that is an eigenvector.(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 from part (d) and the generalized eigenvector in the other two columns. Find T1 andform the product J = T1AT. The matrix J is the Jordan form of A.