1. Prove that the following statements are equivalent: (i) (A,B) is controllable; (ii) [s1 - A B] has full rank for all complex numbers s; (iii) The controllability matrix e = [B AB A'B] has full rank; and (iv) ["BB"et" is positive definite for all T>0, wherein Ae**** and Be*** 1. Prove that the following statements are equivalent: (i) (A,B) is controllable; (ii) [s1 - A B] has full rank for all complex numbers s; (iii) The controllability matrix e = [B AB A'B] has full rank; and (iv) ["BB"et" is positive definite for all T>0, wherein Ae**** and Be***