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.

 

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Q4. Let A be a complex matrix with minimal polynomial x²(x − 1)². - (i) Let h(z) = z", n ≥ 4. Find a degree 3 real polynomial f(z) such that ƒ(1) = h(1), ƒ'(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 A² + √n A³ for some real numbers an, n, Yn and Sn. Q4. Let A be a complex matrix with minimal polynomial x²(x − 1)². - (i) Let h(z) = z", n ≥ 4. Find a degree 3 real polynomial f(z) such that ƒ(1) = h(1), ƒ'(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 A² + √n A³ for some real numbers an, n, Yn and Sn.

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A be a complex matrix Minimal polynomial of A mx xx1 i hzz n 4 Sinc... View the full answer