Minimal polynomial and stabilizer

Related BrainMass Solutions

• Give an example of an operator on C^3 whose minimal polynomial is z^2.

  A minimal polynomial of an operator A is a minimal p(z) such that p(A) = 0.

• Determine the minimal polynomial over Q for the element 1+i.

  115756 Determining the Minimal Polynomial Determine the minimal polynomial over Q for the element 1+i. The minimal polynomial of z=1+i is the monic polynomial with rational coefficients, of least degree, having z as a root.

• Minimal Polynomial of T

  440664 Minimal Polynomial of T Let V be an inner-product space and suppose T element of L(V) is normal. a) Prove that null T^k = null T for all positive integer k. b) Prove that the minimal polynomial of T has no repeated roots.

• Linear operators

  Then we find a polynomial with a degree less than . This implies that is not the minimal polynomial. We get a contradiction. Therefore, there must exist a vector such that .

• Linear Algebra : Diagonalizing Matrices

  Then the minimal polynomial (let's call it m(x)) will definitely divide f(x)=x^2-x=x(x-1).

• Problems in Galois Theory

  Thus we see that is a root of the polynomial f(x) = x^4 - 10x^2 +1. We claim that f is the minimal polynomial of alpha. If not, then the minimal polynomial of would have degree 2 or less.

• Find Irreducible Polynomial

  Therefore the minimal polinom of K over Q is the minimal polinom of a which is x^4-16x^2+4. An irreducible polynomial defining an extension is found.

• Divisibility proofs

  This provides an example of divisibility proofs with a minimal polynomial.

• Let T be an element of L ( complex numbers^3 ) and suppose 0 is the only eigenvalue for T. Show that T^n = 0 for some n.

  Now we consider the minimal polynomial m(x) of T. We know that m(x) divides f(x) = x^3, then m(x) = x^n for some n. Since m(x) is the minimal polynomial of T, then m(T) = 0. This implies that m(T) = T^n = 0 for some n. Done.