Let A be a real 2 ( 2 matrix with complex eigenvalues ‹‹ = a ( bi such that b ‰  0 and ( ‹‹ ( = 1. Prove that every trajectory of the dynamical system xk+1 = Axk lies on an ellipse. [Hint: Theorem 4.43 shows that if v is an eigenvector corresponding to ‹‹ = a - bi, then the matrix P = [Re v Im v] is invertible and

Show that the quadratic xT Bx = k defines an ellipse for all k > 0, and prove that if x lies on this ellipse, so does Ax.]

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-t, P-1-Set B = (PP*)-1 a A=P