Find the Eigenvectors/Eigenspace

Problem

[[2,0,0,0],[0,1,0,1],[0,0,2,0],[0,1,0,1]]

Solution

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

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

2. Calculate the determinant of A−λ*I by expanding along the first row.

det(2−λ)=(2−λ)*det(1−λ)

3. Expand the remaining 3×3 determinant along its second row.

(2−λ)*(2−λ)*det(1−λ)=(2−λ)2*((1−λ)2−1)

4. Solve for λ by setting the characteristic polynomial to zero.

(2−λ)2*(λ2−2*λ)=0

(2−λ)2*λ*(λ−2)=0

(λ_1)=2,(λ_2)=0

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

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

−(x_2)+(x_4)=0⇒(x_2)=(x_4)

(E_2)=span{[[1],[0],[0],[0]],[[0],[0],[1],[0]],[[0],[1],[0],[1]]}

6. Find the eigenspace for λ=0 by solving A*v=0

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

2*(x_1)=0,2*(x_3)=0,(x_2)+(x_4)=0⇒(x_1)=0,(x_3)=0,(x_2)=−(x_4)

(E_0)=span{0}

Final Answer

(E_2)=span{[[1],[0],[0],[0]],[[0],[0],[1],[0]],[[0],[1],[0],[1]]},(E_0)=span{0}

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