Find the Eigenvectors/Eigenspace [[-16,10],[-20,14]]

Problem

[[−16,10],[−20,14]]

Solution

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

A=[[−16,10],[−20,14]]

2. Set up the determinant for the characteristic polynomial.

det(−16−λ)=0

3. Expand the determinant to form a quadratic equation.

(−16−λ)*(14−λ)−(10)*(−20)=0

λ2+2*λ−224+200=0

λ2+2*λ−24=0

4. Factor the quadratic equation to find the eigenvalues (λ_1) and (λ_2)

(λ+6)*(λ−4)=0

(λ_1)=−6,(λ_2)=4

5. Find the eigenvector for (λ_1)=−6 by solving (A−(−6)*I)*v=0

[[−16+6,10],[−20,14+6]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[−10,10],[−20,20]]*[[(x_1)],[(x_2)]]=[[0],[0]]

−10*(x_1)+10*(x_2)=0⇒(x_1)=(x_2)

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

6. Find the eigenvector for (λ_2)=4 by solving (A−4*I)*v=0

[[−16−4,10],[−20,14−4]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[−20,10],[−20,10]]*[[(x_1)],[(x_2)]]=[[0],[0]]

−20*(x_1)+10*(x_2)=0⇒2*(x_1)=(x_2)

(v_2)=[[1],[2]]

Final Answer

Eigenspaces: *(E_−6)=span*{[1],[1]},(E_4)=span*{[1],[2]}

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