Find the Eigenvectors/Eigenspace

Problem

A=[[2,0,0,1],[1,2,0,−1],[1,5,−2,−1],[0,0,0,2]]

Solution

1. Set up the characteristic equation by calculating det(A−λ*I)=0 to find the eigenvalues.

det(2−λ)=0

2. Expand the determinant along the bottom row.

(2−λ)*det(2−λ)=0

3. Solve for eigenvalues by expanding the remaining 3×3 triangular determinant.

(2−λ)*(2−λ)*(2−λ)*(−2−λ)=0

(λ_1)=2,(λ_2)=−2

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

[[0,0,0,1],[1,0,0,−1],[1,5,−4,−1],[0,0,0,0]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

5. Solve the system for λ=2 From row 1, (x_4)=0 From row 2, (x_1)−(x_4)=0⇒(x_1)=0 From row 3, 5*(x_2)−4*(x_3)=0⇒(x_2)=4/5*(x_3)

(E_2)=span*{[0],[4],[5],[0]}

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

[[4,0,0,1],[1,4,0,−1],[1,5,0,−1],[0,0,0,4]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

7. Solve the system for λ=−2 From row 4, (x_4)=0 From row 1, 4*(x_1)=0⇒(x_1)=0 From row 2, 4*(x_2)=0⇒(x_2)=0 (x_3) is a free variable.

(E_−2)=span*{[0],[0],[1],[0]}

Final Answer

(E_2)=span*{[0],[4],[5],[0]},(E_−2)=span*{[0],[0],[1],[0]}

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