Find the Eigenvectors/Eigenspace [[-16,8],[8,-4]]

Problem

[[−16,8],[8,−4]]

Solution

1. Find the characteristic equation by setting the determinant of A−λ*I to zero.

det(−16−λ)=0

2. Expand the determinant to form a quadratic equation in terms of λ

(−16−λ)*(−4−λ)−(8)*(8)=0

λ2+20*λ+64−64=0

λ2+20*λ=0

3. Solve for the eigenvalues by factoring the characteristic polynomial.

λ*(λ+20)=0

(λ_1)=0,(λ_2)=−20

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

[[−16,8],[8,−4]]*[[(x_1)],[(x_2)]]=[[0],[0]]

−16*(x_1)+8*(x_2)=0⇒2*(x_1)=(x_2)

(v_1)=[[1],[2]]

5. Find the eigenvector for (λ_2)=−20 by solving (A+20*I)*v=0

[[−16+20,8],[8,−4+20]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[4,8],[8,16]]*[[(x_1)],[(x_2)]]=[[0],[0]]

4*(x_1)+8*(x_2)=0⇒(x_1)=−2*(x_2)

(v_2)=[[−2],[1]]

Final Answer

(E_0)=span{1},(E_−20)=span{−2}

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