Find the Eigenvectors/Eigenspace [[1,1],[1,1]]

Problem

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

Solution

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

A=[[1,1],[1,1]]

det(1−λ)=0

2. Solve the characteristic polynomial for λ

(1−λ)2−1=0

λ2−2*λ+1−1=0

λ*(λ−2)=0

(λ_1)=0,(λ_2)=2

3. Find the eigenvector for (λ_1)=0 by solving (A−0*I)*v=0

[[1,1],[1,1]]*[[x],[y]]=[[0],[0]]

x+y=0⇒y=−x

(v_1)=[[1],[−1]]

4. Find the eigenvector for (λ_2)=2 by solving (A−2*I)*v=0

[[1−2,1],[1,1−2]]*[[x],[y]]=[[0],[0]]

[[−1,1],[1,−1]]*[[x],[y]]=[[0],[0]]

−x+y=0⇒y=x

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

5. Define the eigenspaces (E_λ) as the span of the calculated eigenvectors.

(E_0)=span{1}

(E_2)=span{1}

Final Answer

Eigenspaces: *(E_0)=span{1},(E_2)=span{1}

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