Find the Eigenvectors/Eigenspace [[-4,6],[8,-6]]

Problem

[[−4,6],[8,−6]]

Solution

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

A=[[−4,6],[8,−6]]

det(−4−λ)=0

2. Expand the determinant to form the characteristic polynomial.

(−4−λ)*(−6−λ)−(6)*(8)=0

λ2+10*λ+24−48=0

λ2+10*λ−24=0

3. Factor the quadratic equation to solve for the eigenvalues (λ_1) and (λ_2)

(λ+12)*(λ−2)=0

(λ_1)=−12,(λ_2)=2

4. Find the eigenvector for (λ_1)=−12 by solving (A−(−12)*I)*v=0

[[−4+12,6],[8,−6+12]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[8,6],[8,6]]*[[(x_1)],[(x_2)]]=[[0],[0]]

8*(x_1)+6*(x_2)=0⇒4*(x_1)=−3*(x_2)

(v_1)=[[−3],[4]]

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

[[−4−2,6],[8,−6−2]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[−6,6],[8,−8]]*[[(x_1)],[(x_2)]]=[[0],[0]]

−6*(x_1)+6*(x_2)=0⇒(x_1)=(x_2)

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

Final Answer

(E_−12)=span*{[−3],[4]},(E_2)=span*{[1],[1]}

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