In this problem we illustrate how a 2 2 system with eigenvalues i can betransformed into
Chapter 9, Problem 22(choose chapter or problem)
In this problem we illustrate how a 2 2 system with eigenvalues i can betransformed into the system (11). Consider the system in 12:x = 2 2.51.8 1x = Ax. (i)(a) Show that the eigenvalues of this system are r1 = 0.5 + 1.5i and r2 = 0.5 1.5i.(b) Show that the eigenvector corresponding to r1 can be chosen as(1) = 53 3i=53+ i 03. (ii)(c) Let P be the matrix whose columns are the real and imaginary parts of (1). ThusP =5 03 3. (iii)Let x = Py and substitute for x in Eq. (i). Show thaty = (P1AP)y. (iv)(d) Find P1 and show thatP1AP = 0.5 1.51.5 0.5. (v)Thus Eq. (v) has the form of Eq. (11).
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