Find the Eigenvectors/Eigenspace [[5,3],[3,5]]

Problem

[[5,3],[3,5]]

Solution

1. Identify the matrix A and the characteristic equation det(A−λ*I)=0 to find the eigenvalues.

A=[[5,3],[3,5]]

det(5−λ)=0

2. Expand the determinant to form the characteristic polynomial.

(5−λ)*(5−λ)−(3)*(3)=0

λ2−10*λ+25−9=0

λ2−10*λ+16=0

3. Solve the quadratic equation for λ by factoring.

(λ−8)*(λ−2)=0

(λ_1)=8,(λ_2)=2

4. Find the eigenspace for (λ_1)=8 by solving (A−8*I)*v=0

[[5−8,3],[3,5−8]]*[[x],[y]]=[[0],[0]]

[[−3,3],[3,−3]]*[[x],[y]]=[[0],[0]]

−3*x+3*y=0⇒x=y

(E_8)=span*{[1],[1]}

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

[[5−2,3],[3,5−2]]*[[x],[y]]=[[0],[0]]

[[3,3],[3,3]]*[[x],[y]]=[[0],[0]]

3*x+3*y=0⇒x=−y

(E_2)=span*{[1],[−1]}

Final Answer

(E_8)=span*{[1],[1]},(E_2)=span*{[1],[−1]}

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