Find the Eigenvectors/Eigenspace

Problem

A=[[5,5,5,5],[6,5,5,6],[7,8,9,8],[6,7,8,9]]

Solution

1. Find the eigenvalues by solving the characteristic equation det(A−λ*I)=0 For this specific matrix, the eigenvalues are (λ_1)=26 (λ_2)=2 (λ_3)=0 and (λ_4)=0

2. Find the eigenvector for λ=26 by solving (A−26*I)*v=0 This leads to the system:

[[−21,5,5,5],[6,−21,5,6],[7,8,−17,8],[6,7,8,−17]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

Row reduction yields the eigenvector (v_1)=[[5],[6],[8],[8]]

3. Find the eigenvector for λ=2 by solving (A−2*I)*v=0 This leads to the system:

[[3,5,5,5],[6,3,5,6],[7,8,7,8],[6,7,8,7]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

Row reduction yields the eigenvector (v_2)=[[5],[6],[−8],[−8]]

4. Find the eigenspace for λ=0 by solving A*v=0 This leads to the system:

[[5,5,5,5],[6,5,5,6],[7,8,9,8],[6,7,8,9]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

Row reduction results in the reduced row echelon form:

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

This gives two free variables, (x_3) and (x_4) Setting (x_3)=−1,(x_4)=0 gives (v_3)=[[0],[1],[−1],[0]] Setting (x_3)=0,(x_4)=−1 gives (v_4)=[[1],[0],[0],[−1]]

Final Answer

(E_26)=span*{[5],[6],[8],[8]},(E_2)=span*{[5],[6],[−8],[−8]},(E_0)=span*{[[0],[1],[−1],[0]],[[1],[0],[0],[−1]]}

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