Find the Eigenvectors/Eigenspace [[3,-2],[-1,4]]

Problem

[[3,−2],[−1,4]]

Solution

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

A=[[3,−2],[−1,4]]

det(3−λ)=0

2. Expand the determinant to form the characteristic polynomial.

(3−λ)*(4−λ)−(−2)*(−1)=0

λ^2−7*λ+12−2=0

λ^2−7*λ+10=0

3. Factor the quadratic equation to solve for the eigenvalues λ

(λ−2)*(λ−5)=0

(λ_1)=2,(λ_2)=5

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

[[3−2,−2],[−1,4−2]]*[[(x_1)],[(x_2)]]=[[0],[0]]

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

(x_1)−2*(x_2)=0⇒(x_1)=2*(x_2)

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

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

[[3−5,−2],[−1,4−5]]*[[(x_1)],[(x_2)]]=[[0],[0]]

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

−(x_1)−(x_2)=0⇒(x_1)=−(x_2)

(v_2)=[[−1],[1]]

Final Answer

Eigenspaces: *(E_2)=span*{[2],[1]},(E_5)=span*{[−1],[1]}

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