Find the Eigenvectors/Eigenspace [[2,1,1],[1,2,1],[1,1,2]]

Problem

[[2,1,1],[1,2,1],[1,1,2]]

Solution

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

det(2−λ)=0

2. Expand the determinant to find the characteristic polynomial.

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

(2−λ)*(λ2−4*λ+3)−(1−λ)+(λ−1)=0

(2−λ)*(λ−1)*(λ−3)+2*(λ−1)=0

3. Factor the polynomial to find the eigenvalues.

(λ−1)*((2−λ)*(λ−3)+2)=0

(λ−1)*(−λ2+5*λ−6+2)=0

−(λ−1)*(λ2−5*λ+4)=0

−(λ−1)*(λ−1)*(λ−4)=0

(λ_1)=4,(λ_2)=1,(λ_3)=1

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

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

Row reduction leads to *x=z,y=z

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

5. Find the eigenspace for λ=1 by solving (A−1*I)*v=0

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

x+y+z=0⇒x=−y−z

v=[[−y−z],[y],[z]]=y*[[−1],[1],[0]]+z*[[−1],[0],[1]]

Final Answer

(E_4)=span*{[1],[1],[1]},(E_1)=span*{[[−1],[1],[0]],[[−1],[0],[1]]}

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