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

Problem

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

Solution

1. Identify the matrix A and the identity matrix I

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

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

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

det(1−λ)=0

(1−λ)2−(1)*(0)=0

(1−λ)2=0

λ=1

3. Set up the system for the eigenvector v=[[x],[y]] using the eigenvalue λ=1 in the equation (A−λ*I)*v=0

[[1−1,1],[0,1−1]]*[[x],[y]]=[[0],[0]]

[[0,1],[0,0]]*[[x],[y]]=[[0],[0]]

4. Solve the system of linear equations derived from the matrix multiplication.

0*x+1*y=0

y=0

5. Determine the eigenspace by expressing the solution in terms of the free variable x

v=[[x],[0]]=x*[[1],[0]]

Final Answer

(E_1)=span*{[1],[0]}

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