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

Problem

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

Solution

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

det(1−λ)=0

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

(1−λ)*(2−λ)−(3)*(4)=0

λ2−3*λ+2−12=0

λ2−3*λ−10=0

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

(λ−5)*(λ+2)=0

(λ_1)=5,(λ_2)=−2

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

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

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

−4*(x_1)+3*(x_2)=0⇒(x_2)=4/3*(x_1)

(v_1)=[[3],[4]]

5. Find the eigenvector for (λ_2)=−2 by solving the homogeneous system (A+2*I)*v=0

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

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

3*(x_1)+3*(x_2)=0⇒(x_2)=−(x_1)

(v_2)=[[1],[−1]]

Final Answer

(E_5)=span*{[3],[4]},(E_−2)=span*{[1],[−1]}

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