Let A be a real, square matrix. Suppose it admits a complex eigenvalue λ = a + ib, where a, b = R. (a) By making reference to the characteristic polynomial of A, explain why = a - ib must also be an eigenvalue. Furthermore, explain why the corresponding eigenvector æ of A is necessarily complex. (b) Decompose the eigenvector x into real and imaginary parts as x = u+iv. Show that y = u - iv is an eigenvector for X. (c) Suppose from now on that A is a 2 x 2-matrix and that u and v are independent. With respect to the basis {u, v} show that A is (d) The differential equation ż= Az admits two independent complex solutions Z₁ (t) = etx, and z2(t) = exty. By splitting these complex solutions into their real and imaginary parts obtain two independent real solutions to the differential equation. Your solutions should be written in terms of u, v and a, b. Screenshot