Find the Eigenvectors/Eigenspace

Problem

[[6,−3,1,0],[0,3,1,0],[−6,6,0,0],[3,3,−2,3]]

Solution

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

det(6−λ)=0

2. Expand the determinant along the last column.

(3−λ)*det(6−λ)=0

3. Solve for the eigenvalues λ by expanding the 3×3 determinant.

(3−λ)*[(6−λ)*((3−λ)*(−λ)−6)−0+(−6)*(−3−(3−λ))]=0

(3−λ)*[(6−λ)*(λ2−3*λ−6)−6*(λ−6)]=0

(3−λ)*[−λ3+9*λ2−12*λ−36−6*λ+36]=0

(3−λ)*(−λ3+9*λ2−18*λ)=0

−(3−λ)*(λ)*(λ2−9*λ+18)=0

−(3−λ)*(λ)*(λ−3)*(λ−6)=0

(λ_1)=0,(λ_2)=3,(λ_3)=6

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

[[6,−3,1,0],[0,3,1,0],[−6,6,0,0],[3,3,−2,3]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

Row reduction leads to (x_1)=(x_2) (x_3)=−3*(x_2) and (x_4)=−3*(x_2)

(E_0)=span*{[1],[1],[−3],[−3]}

5. Find the eigenspace for λ=3 by solving (A−3*I)*v=0

[[3,−3,1,0],[0,0,1,0],[−6,6,−3,0],[3,3,−2,0]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

From row 2, (x_3)=0 Then 3*(x_1)−3*(x_2)=0⇒(x_1)=(x_2) The variable (x_4) is free.

(E_3)=span*{[[1],[1],[0],[0]],[[0],[0],[0],[1]]}

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

[[0,−3,1,0],[0,−3,1,0],[−6,6,−6,0],[3,3,−2,−3]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

From row 1, (x_3)=3*(x_2) Substituting into row 3: −6*(x_1)+6*(x_2)−18*(x_2)=0⇒(x_1)=−2*(x_2) Substituting into row 4: 3*(−2*(x_2))+3*(x_2)−2*(3*(x_2))−3*(x_4)=0⇒(x_4)=−3*(x_2)

(E_6)=span*{[−2],[1],[3],[−3]}

Final Answer

(E_0)=span*{[1],[1],[−3],[−3]},(E_3)=span*{[[1],[1],[0],[0]],[[0],[0],[0],[1]]},(E_6)=span*{[−2],[1],[3],[−3]}

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