Find the Eigenvectors/Eigenspace [[-2,-1,-1],[-2,-3,-2],[2,2,1]]

Problem

[[−2,−1,−1],[−2,−3,−2],[2,2,1]]

Solution

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

det(−2−λ)=0

2. Calculate the determinant using cofactor expansion along the first row.

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

3. Simplify the polynomial to find the eigenvalues.

−(λ+1)2*(λ+2)=0

(λ_1)=−1,(λ_2)=−2

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

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

5. Row reduce the matrix to find the basis for the eigenspace.

[[1,1,1],[0,0,0],[0,0,0]]

x+y+z=0⇒x=−y−z

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

6. Find the eigenspace for λ=−2 by solving (A+2*I)*v=0

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

7. Row reduce the matrix for the second eigenvalue.

[[1,0,1/2],[0,1,1],[0,0,0]]

x=−1/2*z,y=−z

(E_−2)=span*{[−1],[−2],[2]}

Final Answer

(E_−1)=span*{[[−1],[1],[0]],[[−1],[0],[1]]},(E_−2)=span*{[−1],[−2],[2]}

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