Find the Eigenvectors/Eigenspace [[3,1,1],[1,3,1],[1,1,3]]

Problem

[[3,1,1],[1,3,1],[1,1,3]]

Solution

1. Identify the matrix A and the identity matrix I

A=[[3,1,1],[1,3,1],[1,1,3]]

I=[[1,0,0],[0,1,0],[0,0,1]]

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

det(3−λ)=0

(3−λ)*((3−λ)2−1)−1*((3−λ)−1)+1*(1−(3−λ))=0

(3−λ)*(λ2−6*λ+8)−(2−λ)+(λ−2)=0

(3−λ)*(λ−2)*(λ−4)+2*(λ−2)=0

(λ−2)*((3−λ)*(λ−4)+2)=0

(λ−2)*(−λ2+7*λ−10)=0

−(λ−2)*(λ−2)*(λ−5)=0

(λ_1)=5,(λ_2)=2,(λ_3)=2

3. Solve for the eigenspace of λ=5 by solving (A−5*I)*v=0

[[−2,1,1],[1,−2,1],[1,1,−2]]*[[x],[y],[z]]=[[0],[0],[0]]

Row reduction leads to x=z and y=z

(E_5)=span*{[1],[1],[1]}

4. Solve for the eigenspace of λ=2 by solving (A−2*I)*v=0

[[1,1,1],[1,1,1],[1,1,1]]*[[x],[y],[z]]=[[0],[0],[0]]

This results in the single equation x+y+z=0 or x=−y−z

(E_2)=span*{[[−1],[1],[0]],[[−1],[0],[1]]}

Final Answer

(E_5)=span*{[1],[1],[1]},(E_2)=span*{[[−1],[1],[0]],[[−1],[0],[1]]}

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