Let A be a complex matrix with minimal polynomial x^2*(x − 1)^2 .

(i) Let h(z) = z^n , n ≥ 4. Find a degree 3 real polynomial f(z) such that f(1) = h(1), f' (1) = h'(1), f(0) = h(0), f'(0) = h'(0).

(ii) Derive a formula for the power A^n in the form A^n = αnI + βnA + γnA^2 + δnA^3 for some real numbers αn, βn, γn and δn.

 

Q4. Let A be a complex matrix with minimal polynomial x(x 1). - (i) Let h(z) = z", n 4. Find a degree 3 real polynomial f(z) such that (1) = h(1), '(1) = h'(1), f(0) = h(0), f'(0) = h'(0). (ii) Derive a formula for the power An in the form An = anI+BnA + n A + n A for some real numbers an, n, Yn and Sn.