Find the Eigenvectors/Eigenspace A=[[6,-3],[-2,1]]

Problem

A=[[6,−3],[−2,1]]

Solution

1. Find the characteristic equation by calculating det(A−λ*I)=0

det(6−λ)=0

2. Expand the determinant to find the characteristic polynomial.

(6−λ)*(1−λ)−(−3)*(−2)=0

λ2−7*λ+6−6=0

λ2−7*λ=0

3. Solve for the eigenvalues by factoring the polynomial.

λ*(λ−7)=0

(λ_1)=0,(λ_2)=7

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

[[6,−3],[−2,1]]*[[(x_1)],[(x_2)]]=[[0],[0]]

−2*(x_1)+(x_2)=0⇒(x_2)=2*(x_1)

(v_1)=[[1],[2]]

5. Find the eigenvector for (λ_2)=7 by solving (A−7*I)*v=0

[[6−7,−3],[−2,1−7]]=[[−1,−3],[−2,−6]]

−(x_1)−3*(x_2)=0⇒(x_1)=−3*(x_2)

(v_2)=[[−3],[1]]

Final Answer

Eigenspaces: *(E_0)=span*{[1],[2]},(E_7)=span*{[−3],[1]}

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