Solved: The two parts of this exercise lead you through a proof of Theorem 5.3.8. (a)

Chapter 5, Problem 35

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The two parts of this exercise lead you through a proof of Theorem 5.3.8. (a) For notational simplicity, let M = a b b a and let u = Re(x) and v = Im(x), so P = [u | v]. Show that the relationship Ax = x implies that Ax = (au + bv) + i(bu + av) and then equate real and imaginary parts in this equation to show that AP = [Au | Av]=[au + bv | bu + av] = PM (b) Show that P is invertible, thereby completing the proof, since the result in part (a) implies that A = PMP 1. [Hint: If P is not invertible, then one of its column vectors is a real scalar multiple of the other, say v = cu. Substitute this into the equations Au = au + bv and Av = bu + av obtained in part (a), and show that (1 + c2)bu = 0. Finally, show that this leads to a contradiction, thereby proving that P is invertible.]