Consider the motion of a pair of undamped harmonic oscillators, given by the equations Page 2 of 3 x= -x1, x= -wx2, W1, W2 ER\{0}. Math 119a The location of the first oscillator is given by x1(t), and the location of the second oscillator is given by x2(t). (a) Introduce variables y₁, y2 for the velocities of each oscillators and formulate the motion of the harmonic oscillators as a four dimensional linear system of the form X' = AX, AЄ M₁(R), X = (x1,y1, x2, y2). (b) Find the eigenvalues and eigenvectors of A. (c) Show there exists TЄ M₁(R) invertible so that T-1AT = B, where B is in Jordan canonical form. (d) Use the Jordan canonical form to find the general solution of the system X' = AX. Justify your answer. You may use the following two facts: a cos b sin b FACT 1: if C = for a, b R, then e = ea -b 9 sinb cos b (8) [M₁ 0 0 0 0 M2 0 0 FACT 2: if M is a block diagonal matrix M = (9) 0 0 0 0 0 0 Mk [MP 0 0 0 0 M 0 0 then MP = for all p = 1,2,3,.... (10) 0 0 0 0 0 0 M