plz write the detail and also tell method of work ASAP

No.1

Let S be an endomorphism of a finite dimensional vector space V over a field F whose characteristic polynomial is not equal to its minimal polynomial. Show that there is an endomorphism T of V so that T commutes with S but T is not a polynomial in S (T is a polynomial in S if T = aol taj StayS'+. . .+a,S* for some k and a; ( F.)Let A be an n x n real matrix, all of whose (complex) eigenvalues are real and positive. Show that for any integer m 2 1, there exists at least one n x n real matrix B with Bm = A. Hint. Make use of the Jordan form S + N of a conjugate of A, where S is diagonal and N is strictly triangular.Let V be a finite dimensional vector space over an algebraically closed field K, and let T : V - V be a linear transformation. (a) Show that there are T-invariant subspaces V, C V and elements of a; E K such that V is the direct sum of the V, and (T - o;l) : W, - V. is nilpotent. (A transformation N is nilpotent if N" = 0, for some n > 1.) (b) show that there are polynomials S(T), N(T) ( K [T] such that T = S(T) + N(T), S(T) : V - V is diagonalizable, and N(T) : V - V is nilpotent. Hint. Use the chinese Remainder Theorem