Reference no: EM132435799

Task 1:

If p(T) = Tⁿ- α_n-1 T^n-1- α_n-2 T^n-2- .........- α₀∈K[T] is a polynomial

Then this matrix is the companion matrix of p:

Show that:

1) For the characteristics polynomial X_A(T) of p applies X_A(T)  = (-λ)ⁿp(T).

2) if λ is the roof of p(t), so is (λ, λ, .... λ^n-1)^T the eigenvector of A to the eigen value λ.

3) The polynomial p(t) has different roots λ₁.... λ_n ∈ k.   

Define a matrix S, so that S^-1AS a diagonal matrix is

Task 2:

If V is a n- dimensional vectorspace and f:V→V be a diagonalizable endomorphism.

Show that : If E ⊆ V is a f- invariant vector subspace (which means if f(x) ∈ E for all x ∈ E applies, then the restriction of f|_E is diagonalizable.

Tip 1: Create a basis x₁.......x_k, y_k+1,......y_n of V so that E = L (x₁......,x_k) is and y_k+1,.....y_n eigenvectors of f are.

Tip 2: If W = L(y_k+1, ..... y_n) so is V = E⊕W, then write fitting vectored y₁,... y_k as u_i + ω_i and show that u_i is linear independent.