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

Problem

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

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)*(4−λ)−(1)*(3)=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 (A−5*I)*v=0

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

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

−3*(x_1)+(x_2)=0⇒(x_2)=3*(x_1)

(v_1)=[[1],[3]]

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

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

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

(x_1)+(x_2)=0⇒(x_2)=−(x_1)

(v_2)=[[1],[−1]]

Final Answer

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

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