Find the Eigenvectors/Eigenspace [[8,6],[-3,2]]

Problem

[[8,6],[−3,2]]

Solution

1. Identify the matrix A and the characteristic equation det(A−λ*I)=0 to find the eigenvalues.

A=[[8,6],[−3,2]]

2. Set up the determinant for the characteristic polynomial.

det(8−λ)=0

3. Expand the determinant to form a quadratic equation.

(8−λ)*(2−λ)−(6)*(−3)=0

λ2−10*λ+16+18=0

λ2−10*λ+34=0

4. Solve for λ using the quadratic formula λ=(−b±√(,b2−4*a*c))/(2*a)

λ=(10±√(,(−10)2−4*(1)*(34)))/2

λ=(10±√(,100−136))/2

λ=(10±√(,−36))/2

λ=5±3*i

5. Find the eigenvector for (λ_1)=5+3*i by solving (A−(λ_1)*I)*v=0

[[8−(5+3*i),6],[−3,2−(5+3*i)]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[3−3*i,6],[−3,−3−3*i]]*[[(x_1)],[(x_2)]]=[[0],[0]]

6. Solve the system of equations. From the second row:

−3*(x_1)+(−3−3*i)*(x_2)=0

−3*(x_1)=(3+3*i)*(x_2)

(x_1)=(−1−i)*(x_2)

7. Determine the eigenspace (E_(λ_1)) by choosing (x_2)=1

(v_1)=[[−1−i],[1]]

8. Find the eigenvector for the conjugate eigenvalue (λ_2)=5−3*i by taking the conjugate of (v_1)

(v_2)=[[−1+i],[1]]

Final Answer

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

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