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

Problem

[[2,3],[1,4]]

Solution

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

det(2−λ)=0

2. Expand the determinant to form a quadratic equation in terms of λ

(2−λ)*(4−λ)−(3)*(1)=0

λ2−6*λ+8−3=0

λ2−6*λ+5=0

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

(λ−5)*(λ−1)=0

(λ_1)=5,(λ_2)=1

4. Find the eigenvector for (λ_1)=5 by solving the system (A−5*I)*v=0

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

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

−3*(x_1)+3*(x_2)=0⇒(x_1)=(x_2)

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

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

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

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

(x_1)+3*(x_2)=0⇒(x_1)=−3*(x_2)

(v_2)=[[−3],[1]]

Final Answer

Eigenspaces: *(E_5)=span*{[1],[1]},(E_1)=span*{[−3],[1]}

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