Find the Eigenvectors/Eigenspace [[1,0,1],[0,-1,0],[2,1,-1]]

Problem

[[1,0,1],[0,−1,0],[2,1,−1]]

Solution

1. Identify the matrix A and set up the characteristic equation det(A−λ*I)=0 to find the eigenvalues.

A=[[1,0,1],[0,−1,0],[2,1,−1]]

det(1−λ)=0

2. Calculate the determinant by expanding along the second row.

(−1−λ)⋅det(1−λ)=0

−(1+λ)*((1−λ)*(−1−λ)−2)=0

−(1+λ)*(λ2−1−2)=0

−(1+λ)*(λ2−3)=0

3. Solve for the eigenvalues λ

(λ_1)=−1

(λ_2)=√(,3)

(λ_3)=−√(,3)

4. Find the eigenvector for (λ_1)=−1 by solving (A+I)*v=0

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

2*x+z=0⇒z=−2*x

2*x+y=0⇒y=−2*x

(v_1)=[[1],[−2],[−2]]

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

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

(−1−√(,3))*y=0⇒y=0

(1−√(,3))*x+z=0⇒z=(√(,3)−1)*x

(v_2)=[[1],[0],[√(,3)−1]]

6. Find the eigenvector for (λ_3)=−√(,3) by solving (A+√(,3)*I)*v=0

[[1+√(,3),0,1],[0,−1+√(,3),0],[2,1,−1+√(,3)]]*[[x],[y],[z]]=[[0],[0],[0]]

(−1+√(,3))*y=0⇒y=0

(1+√(,3))*x+z=0⇒z=(−1−√(,3))*x

(v_3)=[[1],[0],[−1−√(,3)]]

Final Answer

(E_−1)=span*{[1],[−2],[−2]},(E_√(,3))=span*{[1],[0],[√(,3)−1]},(E_−√(,3))=span*{[1],[0],[−1−√(,3)]}

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