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

Problem

A=[[2,−1],[−2,3]]

Solution

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

det(2−λ)=0

2. Expand the determinant to form a quadratic polynomial.

(2−λ)*(3−λ)−(−1)*(−2)=0

λ2−5*λ+6−2=0

λ2−5*λ+4=0

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

(λ−4)*(λ−1)=0

(λ_1)=4,(λ_2)=1

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

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

[[−2,−1],[−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)=1 by solving (A−1*I)*v=0

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

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

(x_1)−(x_2)=0⇒(x_1)=(x_2)

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

Final Answer

(λ_1)=4,(E_4)=span*{[1],[−2]};(λ_2)=1,(E_1)=span*{[1],[1]}

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