Find the Eigenvectors/Eigenspace [[1,-4,2],[2,-7,0],[3,-11,2]]

Problem

[[1,−4,2],[2,−7,0],[3,−11,2]]

Solution

1. Find the eigenvalues by solving the characteristic equation det(A−λ*I)=0

det(1−λ)=0

(1−λ)*((−7−λ)*(2−λ)−0)−(−4)*(2*(2−λ)−0)+2*(2*(−11)−3*(−7−λ))=0

(1−λ)*(λ2+5*λ−14)+4*(4−2*λ)+2*(3*λ−1)=0

−λ3−4*λ2+19*λ−14+16−8*λ+6*λ−2=0

−λ3−4*λ2+17*λ−0=0

−λ*(λ2+4*λ−17)=0

(λ_1)=0,(λ_2)=−2+√(,21),(λ_3)=−2−√(,21)

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

[[1,−4,2],[2,−7,0],[3,−11,2]]*[[x],[y],[z]]=[[0],[0],[0]]

(R_2)−2*(R_1)⇒[[1,−4,2],[0,1,−4],[0,1,−4]]

(R_3)−(R_2)⇒[[1,−4,2],[0,1,−4],[0,0,0]]

y=4*z,x=4*(4*z)−2*z=14*z

(v_1)=[[14],[4],[1]]

3. Find the eigenvector for (λ_2)=−2+√(,21) by solving (A−(−2+√(,21))*I)*v=0

[[3−√(,21),−4,2],[2,−5−√(,21),0],[3,−11,4−√(,21)]]*[[x],[y],[z]]=[[0],[0],[0]]

2*x=(5+√(,21))*y⇒x=(5+√(,21))/2*y

(3−√(,21))*((5+√(,21))/2*y)−4*y+2*z=0

(15+3√(,21)−5√(,21)−21)/2*y−4*y+2*z=0

(−3−√(,21))*y−4*y+2*z=0⇒2*z=(7+√(,21))*y

(v_2)=[[5+√(,21)],[2],[7+√(,21)]]

4. Find the eigenvector for (λ_3)=−2−√(,21) by solving (A−(−2−√(,21))*I)*v=0

[[3+√(,21),−4,2],[2,−5+√(,21),0],[3,−11,4+√(,21)]]*[[x],[y],[z]]=[[0],[0],[0]]

2*x=(5−√(,21))*y⇒x=(5−√(,21))/2*y

(3+√(,21))*((5−√(,21))/2*y)−4*y+2*z=0

(15−3√(,21)+5√(,21)−21)/2*y−4*y+2*z=0

(−3+√(,21))*y−4*y+2*z=0⇒2*z=(7−√(,21))*y

(v_3)=[[5−√(,21)],[2],[7−√(,21)]]

Final Answer

(E_0)=span{14},(E_−2+√(,21))=span{5+√(,21)},(E_−2−√(,21))=span{5−√(,21)}

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