Find the Eigenvectors/Eigenspace [[2,-4],[-1,-1]]

Problem

[[2,−4],[−1,−1]]

Solution

1. Identify the matrix A and the characteristic equation det(A−λ*I)=0 to find the eigenvalues.

A=[[2,−4],[−1,−1]]

det(2−λ)=0

2. Calculate the determinant to form the characteristic polynomial.

(2−λ)*(−1−λ)−(−4)*(−1)=0

λ2−λ−2−4=0

λ2−λ−6=0

3. Factor the quadratic equation to solve for the eigenvalues λ

(λ−3)*(λ+2)=0

(λ_1)=3,(λ_2)=−2

4. Find the eigenvector for (λ_1)=3 by solving (A−3*I)*v=0

[[2−3,−4],[−1,−1−3]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[−1,−4],[−1,−4]]*[[(x_1)],[(x_2)]]=[[0],[0]]

−(x_1)−4*(x_2)=0⇒(x_1)=−4*(x_2)

(v_1)=[[−4],[1]]

5. Find the eigenvector for (λ_2)=−2 by solving (A+2*I)*v=0

[[2+2,−4],[−1,−1+2]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[4,−4],[−1,1]]*[[(x_1)],[(x_2)]]=[[0],[0]]

−(x_1)+(x_2)=0⇒(x_1)=(x_2)

(v_2)=[[1],[1]]

Final Answer

(E_3)=span*{[−4],[1]},(E_−2)=span*{[1],[1]}

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