Question

Exercise 1.6 Give all possible Jordan canonical forms of matrices that have characteristic poly- nomial (t + π)4(1-2)3(t + 2i) and minimal polynomial (t + π)2(t-2)(t + 2). Exercise 1.7 Let A = | 0 1-3) , B = | 2 1 -W | . Are A and B similar? 0 0 1

Answer 1

= (t + pi)^4 (t.2)^3 (t + 2i) implies = (t + pi)^n1 (t - 2)^n2(t + 2)^n3 This given information that matrix is of 2 times 2 order Algebraic multiplicity of evaluate (d_1 = -pi) = 4 A .m (lambda_2 = 2) = 3 A .m (lambda_3 = -2i) = 1 Now Given minimal polynomial m .p = (t + pi)^2 (t -2) (t + 2i) = (t + pi)^m1 (t - 2)^m2(t + 2)^m3 Total no of possible Jordon canonical from = pi^3_i = 1 p(ti) where p(ti) = Partition of ti ti = ni - mi So t_1 = n1 - m1 = 4 - 2 = 2 t_2 = n_2 - m_2 = 3 - 1 = 2 t_3 = n_3 - m_3 = 1 - 1 = 0 So number of possible J.c. from = p(t_1) p(t_2) p(t_2) = p(2) p(2) p(0) = 2 times 2 = 4 So all j.c forms are given as (i