Find the Eigenvectors/Eigenspace [[-5,50,0],[-1,11,1],[0,1,1]]

Problem

[[−5,50,0],[−1,11,1],[0,1,1]]

Solution

1. Find the characteristic equation by calculating the determinant of A−λ*I=0

det(−5−λ)=0

2. Expand the determinant along the first row.

(−5−λ)*((11−λ)*(1−λ)−1)−50*(−1*(1−λ)−0)=0

3. Simplify the polynomial to find the eigenvalues.

(−5−λ)*(λ2−12*λ+10)+50−50*λ=0

−λ3+7*λ2+50*λ−50+50−50*λ=0

−λ3+7*λ2=0

λ2*(7−λ)=0

The eigenvalues are (λ_1)=0 (multiplicity 2) and (λ_2)=7

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

[[−5,50,0],[−1,11,1],[0,1,1]]*[[x],[y],[z]]=[[0],[0],[0]]

Row reduction leads to x=10*y and z=−y
Setting y=1 the eigenvector is (v_1)=[[10],[1],[−1]]
The eigenspace (E_0)=span{10}

5. Find the eigenspace for λ=7 by solving (A−7*I)*v=0

[[−12,50,0],[−1,4,1],[0,1,−6]]*[[x],[y],[z]]=[[0],[0],[0]]

From the third row, y=6*z
From the first row, −12*x+50*(6*z)=0⇒−12*x+300*z=0⇒x=25*z
Setting z=1 the eigenvector is (v_2)=[[25],[6],[1]]
The eigenspace (E_7)=span{25}

Final Answer

(E_0)=span{10},(E_7)=span{25}

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