Find the Eigenvectors/Eigenspace

Problem

A=[[4,0,2,0],[3,2,4,0],[2,−3,1,0],[5,−4,2,3]]

Solution

1. Find the characteristic equation by calculating the determinant of A−λ*I Expanding along the last column:

det(A−λ*I)=(3−λ)*det(4−λ)=0

2. Calculate the 3×3 determinant using the rule of Sarrus or cofactor expansion:

(4−λ)*((2−λ)*(1−λ)+12)+2*(−9−2*(2−λ))=0

(4−λ)*(λ2−3*λ+14)+2*(2*λ−13)=0

−λ3+7*λ2−26*λ+56+4*λ−26=0

−λ3+7*λ2−22*λ+30=0

3. Solve for eigenvalues by factoring the polynomial. Testing λ=3

−(3)3+7*(3)2−22*(3)+30=−27+63−66+30=0

Dividing by (λ−3) gives the quadratic −(λ2−4*λ+10) The roots are:

λ=3,λ=2±i√(,6)

Since the task usually implies real eigenspaces, we focus on λ=3

4. Find the eigenvector for λ=3 by solving (A−3*I)*v=0

[[1,0,2,0],[3,−1,4,0],[2,−3,−2,0],[5,−4,2,0]]*[[(x_1)],[(x_2)],[(x_3)],[(x_4)]]=[[0],[0],[0],[0]]

5. Row reduce the matrix to find the relationships between variables:

(R_2)−3*(R_1)⇒[[1,0,2,0],[0,−1,−2,0],[2,−3,−2,0],[5,−4,2,0]]

(R_3)−2*(R_1)⇒[[1,0,2,0],[0,−1,−2,0],[0,−3,−6,0],[5,−4,2,0]]

(R_3)−3*(R_2)⇒[[1,0,2,0],[0,−1,−2,0],[0,0,0,0],[0,0,0,0]]

The equations are (x_1)+2*(x_3)=0 and −(x_2)−2*(x_3)=0

6. Express the eigenspace in terms of free variables (x_3) and (x_4)

(x_1)=−2*(x_3)

(x_2)=−2*(x_3)

v=[[−2*(x_3)],[−2*(x_3)],[(x_3)],[(x_4)]]=(x_3)*[[−2],[−2],[1],[0]]+(x_4)*[[0],[0],[0],[1]]

Final Answer

(E_3)=span*{[[−2],[−2],[1],[0]],[[0],[0],[0],[1]]}

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