Find the Eigenvectors/Eigenspace [[3,0,-1],[2,3,1],[-3,4,5]]

Problem

A=[[3,0,−1],[2,3,1],[−3,4,5]]

Solution

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

det(3−λ)=0

2. Expand the determinant along the first row.

(3−λ)*((3−λ)*(5−λ)−4)−1*(8−(−3)*(3−λ))=0

3. Simplify the polynomial to find the eigenvalues.

(3−λ)*(λ2−8*λ+11)−(8+9−3*λ)=0

−λ3+11*λ2−32*λ+16=0

−(λ−4)2*(λ−1)=0

(λ_1)=1,(λ_2)=4

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

[[2,0,−1],[2,2,1],[−3,4,4]]*[[x],[y],[z]]=[[0],[0],[0]]

Row reduction yields *z=2*x,y=−3/2*x

(v_1)=[[2],[−3],[4]]

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

[[−1,0,−1],[2,−1,1],[−3,4,1]]*[[x],[y],[z]]=[[0],[0],[0]]

From row 1: *x=−z

From row 2: *2*(−z)−y+z=0⇒y=−z

(v_2)=[[−1],[−1],[1]]

Final Answer

(E_1)=span*{[2],[−3],[4]},(E_4)=span*{[−1],[−1],[1]}

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