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

Problem

[[5,0],[1,5]]

Solution

1. Identify the matrix A and the identity matrix I

A=[[5,0],[1,5]]

I=[[1,0],[0,1]]

2. Find the eigenvalues by solving the characteristic equation det(A−λ*I)=0

det(5−λ)=0

(5−λ)*(5−λ)−(0)*(1)=0

(5−λ)2=0

λ=5

3. Set up the system (A−λ*I)*v=0 for the eigenvalue λ=5 to find the eigenvectors v=[[(x_1)],[(x_2)]]

[[5−5,0],[1,5−5]]*[[(x_1)],[(x_2)]]=[[0],[0]]

[[0,0],[1,0]]*[[(x_1)],[(x_2)]]=[[0],[0]]

4. Solve the linear system resulting from the matrix multiplication.

0*(x_1)+0*(x_2)=0

1*(x_1)+0*(x_2)=0

(x_1)=0

5. Determine the eigenspace by expressing the solution in terms of the free variable (x_2)

v=[[0],[(x_2)]]=(x_2)*[[0],[1]]

Final Answer

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

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