Find the Eigenvectors/Eigenspace [[4,10,-8],[1,2,1],[-1,2,3]]
Problem
[[4,10,−8],[1,2,1],[−1,2,3]]
Solution
Find the characteristic equation by calculating the determinant of A−λ*I=0
det(4−λ)=0
Expand the determinant to find the characteristic polynomial.
(4−λ)*((2−λ)*(3−λ)−2)−10*((3−λ)+1)−8*(2+(2−λ))=0
(4−λ)*(λ2−5*λ+4)−10*(4−λ)−8*(4−λ)=0
Factor the polynomial to find the eigenvalues.
(4−λ)*(λ2−5*λ+4−10−8)=0
(4−λ)*(λ2−5*λ−14)=0
(4−λ)*(λ−7)*(λ+2)=0
(λ_1)=4,(λ_2)=7,(λ_3)=−2
Solve for the eigenvector corresponding to (λ_1)=4 by solving (A−4*I)*v=0
[[0,10,−8],[1,−2,1],[−1,2,−1]]*[[x],[y],[z]]=[[0],[0],[0]]
10*y−8*z=0⇒y=4/5*z
x−2*y+z=0⇒x−8/5*z+z=0⇒x=3/5*z
(v_1)=[[3],[4],[5]]
Solve for the eigenvector corresponding to (λ_2)=7 by solving (A−7*I)*v=0
[[−3,10,−8],[1,−5,1],[−1,2,−4]]*[[x],[y],[z]]=[[0],[0],[0]]
x−5*y+z=0⇒x=5*y−z
−3*(5*y−z)+10*y−8*z=0⇒−5*y−5*z=0⇒y=−z
x=5*(−z)−z=−6*z
(v_2)=[[−6],[−1],[1]]
Solve for the eigenvector corresponding to (λ_3)=−2 by solving (A+2*I)*v=0
[[6,10,−8],[1,4,1],[−1,2,5]]*[[x],[y],[z]]=[[0],[0],[0]]
x+4*y+z=0⇒x=−4*y−z
6*(−4*y−z)+10*y−8*z=0⇒−14*y−14*z=0⇒y=−z
x=−4*(−z)−z=3*z
(v_3)=[[3],[−1],[1]]
Final Answer
(E_4)=span*{[3],[4],[5]},(E_7)=span*{[−6],[−1],[1]},(E_−2)=span*{[3],[−1],[1]}
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