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

Problem

[[3,4],[−2,−3]]

Solution

1. Find the characteristic equation by calculating the determinant of A−λ*I and setting it to zero.

det(3−λ)=0

2. Expand the determinant to find the characteristic polynomial.

(3−λ)*(−3−λ)−(4)*(−2)=0

λ2−9+8=0

λ2−1=0

3. Solve for the eigenvalues by factoring the polynomial.

(λ−1)*(λ+1)=0

(λ_1)=1,(λ_2)=−1

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

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

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

2*(x_1)+4*(x_2)=0⇒(x_1)=−2*(x_2)

(v_1)=[[−2],[1]]

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

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

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

4*(x_1)+4*(x_2)=0⇒(x_1)=−(x_2)

(v_2)=[[−1],[1]]

Final Answer

Eigenspaces: *(E_1)=span*{[−2],[1]},(E_−1)=span*{[−1],[1]}

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