Find the Eigenvectors/Eigenspace [[1,1],[-2,4]]

Problem

[[1,1],[−2,4]]

Solution

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

det(1−λ)=0

2. Expand the determinant to find the characteristic polynomial.

(1−λ)*(4−λ)−(1)*(−2)=0

λ2−5*λ+4+2=0

λ2−5*λ+6=0

3. Solve for the eigenvalues by factoring the quadratic equation.

(λ−2)*(λ−3)=0

(λ_1)=2,(λ_2)=3

4. Find the eigenvector for (λ_1)=2 by solving (A−2*I)*v=0

[[1−2,1],[−2,4−2]]*[[x],[y]]=[[0],[0]]

[[−1,1],[−2,2]]*[[x],[y]]=[[0],[0]]

−x+y=0⇒x=y

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

5. Find the eigenvector for (λ_2)=3 by solving (A−3*I)*v=0

[[1−3,1],[−2,4−3]]*[[x],[y]]=[[0],[0]]

[[−2,1],[−2,1]]*[[x],[y]]=[[0],[0]]

−2*x+y=0⇒y=2*x

(v_2)=[[1],[2]]

Final Answer

Eigenspaces: *(E_2)=span*{[1],[1]},(E_3)=span*{[1],[2]}

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