Find the Eigenvectors/Eigenspace

Problem

[[−3,−5,0,0],[−5,−3,0,0],[0,0,−3,−5],[0,0,−5,−3]]

Solution

1. Identify the block structure of the matrix A The matrix is a block diagonal matrix consisting of two identical 2×2 blocks B=[[−3,−5],[−5,−3]]

2. Calculate the eigenvalues of the block B by solving the characteristic equation det(B−λ*I)=0

(−3−λ)*(−3−λ)−(−5)*(−5)=0

λ^2+6*λ+9−25=0

λ^2+6*λ−16=0

(λ+8)*(λ−2)=0

(λ_1)=2,(λ_2)=−8

3. Find the eigenvectors for (λ_1)=2 by solving (B−2*I)*v=0

[[−5,−5],[−5,−5]]*[[(x_1)],[(x_2)]]=[[0],[0]]

−5*(x_1)−5*(x_2)=0⇒(x_1)=−(x_2)

(v_1)=[[1],[−1]]

4. Find the eigenvectors for (λ_2)=−8 by solving (B+8*I)*v=0

[[5,−5],[5,−5]]*[[(x_1)],[(x_2)]]=[[0],[0]]

5*(x_1)−5*(x_2)=0⇒(x_1)=(x_2)

(v_2)=[[1],[1]]

5. Construct the eigenspaces for the 4×4 matrix A using the block structure. Since B appears twice, each eigenvalue has a geometric multiplicity of 2.

6. Define the eigenspace (E_2) for λ=2

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

7. Define the eigenspace (E_−8) for λ=−8

(E_−8)=span*{[[1],[1],[0],[0]],[[0],[0],[1],[1]]}

Final Answer

(E_2)=span*{[[1],[−1],[0],[0]],[[0],[0],[1],[−1]]},(E_−8)=span*{[[1],[1],[0],[0]],[[0],[0],[1],[1]]}

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