Find the Eigenvectors/Eigenspace [[7,0,4],[0,10,0],[4,0,-8]]

Problem

[[7,0,4],[0,10,0],[4,0,−8]]

Solution

1. Set up the characteristic equation by finding the determinant of A−λ*I and setting it to zero.

det(7−λ)=0

2. Expand the determinant along the second row or column to simplify the calculation.

(10−λ)*det(7−λ)=0

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

(10−λ)*((7−λ)*(−8−λ)−16)=0

(10−λ)*(λ2+λ−56−16)=0

(10−λ)*(λ2+λ−72)=0

(10−λ)*(λ+9)*(λ−8)=0

(λ_1)=10,(λ_2)=8,(λ_3)=−9

4. Find the eigenvector for (λ_1)=10 by solving (A−10*I)*v=0

[[−3,0,4],[0,0,0],[4,0,−18]]*[[x],[y],[z]]=[[0],[0],[0]]

−3*x+4*z=0

4*x−18*z=0

x=0,z=0,y=free

(v_1)=[[0],[1],[0]]

5. Find the eigenvector for (λ_2)=8 by solving (A−8*I)*v=0

[[−1,0,4],[0,2,0],[4,0,−16]]*[[x],[y],[z]]=[[0],[0],[0]]

−x+4*z=0⇒x=4*z

2*y=0⇒y=0

(v_2)=[[4],[0],[1]]

6. Find the eigenvector for (λ_3)=−9 by solving (A+9*I)*v=0

[[16,0,4],[0,19,0],[4,0,1]]*[[x],[y],[z]]=[[0],[0],[0]]

4*x+z=0⇒z=−4*x

19*y=0⇒y=0

(v_3)=[[1],[0],[−4]]

Final Answer

(E_10)=span{0},(E_8)=span{4},(E_−9)=span{1}

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