Find the Eigenvectors/Eigenspace [[5,0,1],[-3,1,0],[-3,0,1]]

Problem

[[5,0,1],[−3,1,0],[−3,0,1]]

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 second column.

(1−λ)*det(5−λ)=0

(1−λ)*((5−λ)*(1−λ)−(−3))=0

(1−λ)*(λ2−6*λ+5+3)=0

(1−λ)*(λ2−6*λ+8)=0

3. Solve for the eigenvalues by factoring the quadratic expression.

(1−λ)*(λ−4)*(λ−2)=0

(λ_1)=1,(λ_2)=2,(λ_3)=4

4. Find the eigenvector for (λ_1)=1 by solving (A−1*I)*v=0

[[4,0,1],[−3,0,0],[−3,0,0]]*[[x],[y],[z]]=[[0],[0],[0]]

⇒−3*x=0⇒x=0

⇒4*(0)+0*y+z=0⇒z=0

⇒y=free variable

(E_1)=span{0}

5. Find the eigenvector for (λ_2)=2 by solving (A−2*I)*v=0

[[3,0,1],[−3,−1,0],[−3,0,−1]]*[[x],[y],[z]]=[[0],[0],[0]]

⇒3*x+z=0⇒z=−3*x

⇒−3*x−y=0⇒y=−3*x

(E_2)=span{1}

6. Find the eigenvector for (λ_3)=4 by solving (A−4*I)*v=0

[[1,0,1],[−3,−3,0],[−3,0,−3]]*[[x],[y],[z]]=[[0],[0],[0]]

⇒x+z=0⇒z=−x

⇒−3*x−3*y=0⇒y=−x

(E_4)=span{1}

Final Answer

(λ_1)=1:(E_1)=span{0},(λ_2)=2:(E_2)=span{1},(λ_3)=4:(E_4)=span{1}

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