Find the Eigenvectors/Eigenspace [[1,0,2],[0,5,0],[3,0,2]]

Problem

[[1,0,2],[0,5,0],[3,0,2]]

Solution

1. Set up the characteristic equation by subtracting λ from the diagonal elements and setting the determinant of A−λ*I to zero.

det(1−λ)=0

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

(5−λ)*((1−λ)*(2−λ)−(3)*(2))=0

3. Solve for the eigenvalues by factoring the quadratic expression inside the parentheses.

(5−λ)*(λ2−3*λ−4)=0

(5−λ)*(λ−4)*(λ+1)=0

(λ_1)=5,(λ_2)=4,(λ_3)=−1

4. Find the eigenvector for (λ_1)=5 by solving (A−5*I)*v=0

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

⇒x=0,z=0,y=free

(v_1)=[[0],[1],[0]]

5. Find the eigenvector for (λ_2)=4 by solving (A−4*I)*v=0

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

⇒y=0,3*x=2*z

(v_2)=[[2],[0],[3]]

6. Find the eigenvector for (λ_3)=−1 by solving (A+I)*v=0

[[2,0,2],[0,6,0],[3,0,3]]*[[x],[y],[z]]=[[0],[0],[0]]

⇒y=0,x=−z

(v_3)=[[−1],[0],[1]]

Final Answer

(E_5)=span{0},(E_4)=span{2},(E_−1)=span{−1}

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