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

Problem

[[5,0],[2,3]]

Solution

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

A=[[5,0],[2,3]]

det(5−λ)=0

2. Calculate the determinant to solve for λ

(5−λ)*(3−λ)−(0)*(2)=0

(5−λ)*(3−λ)=0

(λ_1)=5,(λ_2)=3

3. Find the eigenvector for (λ_1)=5 by solving (A−5*I)*v=0

[[5−5,0],[2,3−5]]*[[x],[y]]=[[0],[0]]

[[0,0],[2,−2]]*[[x],[y]]=[[0],[0]]

2*x−2*y=0⇒x=y

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

4. Find the eigenvector for (λ_2)=3 by solving (A−3*I)*v=0

[[5−3,0],[2,3−3]]*[[x],[y]]=[[0],[0]]

[[2,0],[2,0]]*[[x],[y]]=[[0],[0]]

2*x=0⇒x=0

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

Final Answer

Eigenspaces: *(E_5)=span*{[1],[1]},(E_3)=span*{[0],[1]}

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