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

Problem

[[4,−4],[−2,2]]

Solution

1. Set up the characteristic equation by subtracting λ from the diagonal elements and setting the determinant of the resulting matrix to zero.

det(4−λ)=0

2. Expand the determinant to find the characteristic polynomial.

(4−λ)*(2−λ)−(−4)*(−2)=0

λ2−6*λ+8−8=0

λ2−6*λ=0

3. Solve for the eigenvalues by factoring the polynomial.

λ*(λ−6)=0

(λ_1)=0,(λ_2)=6

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

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

4*(x_1)−4*(x_2)=0⇒(x_1)=(x_2)

(v_1)=[[1],[1]]

5. Find the eigenvector for (λ_2)=6 by solving the system (A−6*I)*v=0

[[4−6,−4],[−2,2−6]]*[[(x_1)],[(x_2)]]=[[0],[0]]

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

−2*(x_1)−4*(x_2)=0⇒(x_1)=−2*(x_2)

(v_2)=[[−2],[1]]

Final Answer

(E_λ=0)=span*{[1],[1]},(E_λ=6)=span*{[−2],[1]}

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