How to think of (e)?

Problem 3. Let n E N, and define the 2n x 2n matrix J : On In -In On and the vector space sp(2n) := {KER x2 : JK + KTJ = 0} . (a) Show that det(J) = 1. (b) Show that for every matrix K e sp(2n), the characteristic polynomial fx of K is even (i.e., has the property that fx(-I) = fx(x) for all x). Conclude that if A is an eigenvalue of K, then -A is also an eigenvalue of K. Hint: Use the identity det(Alm - K) = det(J) det(Alan - K). (c) Show that tr(K) = 0 for all K E sp(2n) (d) Show that if 0 is an eigenvalue of K E sp(2n), then almu (0) is even. (e) Given w = (w1, . .., wn) ER" we define the matrix On Ku = _diag(w) di ag (w) On where W1 diag( w) = (i) Check that Kw E sp(2n). (ii) Prove by induction that the characteristic polynomial of Kw is p(X) = II,, (12 + w?), and then find the (complex) eigenvalues of Kw. (iii) Show that J = Kan is diagonalizable over C. (iv) If w; # w; for all i j, is K,, diagonalizable over C