Find the Eigenvectors/Eigenspace [[-4,-8,-6],[-8,6,-8],[8,-7,10]]

Problem

[[−4,−8,−6],[−8,6,−8],[8,−7,10]]

Solution

1. Find the characteristic equation by calculating det(A−λ*I)=0

det(−4−λ)=0

2. Expand the determinant to find the characteristic polynomial.

−(λ−2)*(λ+10)*(λ−20)=0

3. Identify the eigenvalues by solving the polynomial equation.

(λ_1)=2,(λ_2)=−10,(λ_3)=20

4. Find the eigenvector for (λ_1)=2 by solving (A−2*I)*v=0

[[−6,−8,−6],[−8,4,−8],[8,−7,8]]*[[x],[y],[z]]=[[0],[0],[0]]

Row reduction yields *y=0* and *x=−z

(v_1)=[[−1],[0],[1]]

5. Find the eigenvector for (λ_2)=−10 by solving (A+10*I)*v=0

[[6,−8,−6],[−8,16,−8],[8,−7,20]]*[[x],[y],[z]]=[[0],[0],[0]]

Row reduction yields *x=5*z* and *y=3*z

(v_2)=[[5],[3],[1]]

6. Find the eigenvector for (λ_3)=20 by solving (A−20*I)*v=0

[[−24,−8,−6],[−8,−14,−8],[8,−7,−10]]*[[x],[y],[z]]=[[0],[0],[0]]

Row reduction yields *x=1/14*z* and *y=−27/28*z

Scaling by *28* gives *(v_3)=[[2],[−27],[28]]

Final Answer

(E_2)=span*{[−1],[0],[1]},(E_−10)=span*{[5],[3],[1]},(E_20)=span*{[2],[−27],[28]}

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