Question

Please show all steps. Linear Algebra

Bonus: Let A E Mn(C) such that A0 for some positive integer m. Prove that An 0 Hint: What is the minimal polynomial for A?

Answer 1

Let a element mu_0(c) such that a^m = 0 for some positive integer m c a is nipolent matrix given that a satisfies the equation x^m = 0 but we know if f(x) is minimal polynomial of a hen rho(x) divider x^n implies x^m = rho (x) q (x) for som - poly q (x) roots of p(x) q(x) are all zero roots of p(x) are all zero all the eigen values of minimal a are zero and we have all the roots of minimal poly are also roots of characteristic polynomial. implies all roots of characteristic poly are zero characteristic poly of x^n = 0 but a satisfies characteristic polynomal a^n = 0 a^n = 0