Find the Eigenvectors/Eigenspace [[1,0],[6,-1]]

Problem

[[1,0],[6,−1]]

Solution

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

A=[[1,0],[6,−1]]

det(1−λ)=0

2. Calculate the determinant to find the characteristic polynomial.

(1−λ)*(−1−λ)−(0)*(6)=0

λ2−1=0

3. Solve for the eigenvalues λ

(λ_1)=1

(λ_2)=−1

4. Find the eigenvector for (λ_1)=1 by solving (A−I)*v=0

[[1−1,0],[6,−1−1]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[0,0],[6,−2]]*[[(x_1)],[(x_2)]]=[[0],[0]]

6*(x_1)−2*(x_2)=0⇒3*(x_1)=(x_2)

(v_1)=[[1],[3]]

5. Find the eigenvector for (λ_2)=−1 by solving (A+I)*v=0

[[1+1,0],[6,−1+1]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[2,0],[6,0]]*[[(x_1)],[(x_2)]]=[[0],[0]]

2*(x_1)=0⇒(x_1)=0

(v_2)=[[0],[1]]

Final Answer

Eigenspaces: *(E_1)=span*{[1],[3]},(E_−1)=span*{[0],[1]}

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