Find the Eigenvectors/Eigenspace

Problem

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

Solution

1. Identify the eigenvalues of the matrix. Since the matrix is in upper triangular form, the eigenvalues are the diagonal entries.

(λ_1)=1

(λ_2)=0

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

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

3. Solve the system for λ=1 From the rows, we find −5*(x_3)=0 −(x_3)+(x_4)=0 and −(x_4)=0 which implies (x_3)=0 and (x_4)=0 The variables (x_1) and (x_2) are free.

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

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

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

5. Solve the system for λ=0 From the rows, we have (x_1)−5*(x_3)=0 (x_2)=0 and (x_4)=0 Setting the free variable (x_3)=1 we get (x_1)=5

(E_0)=span*{[5],[0],[1],[0]}

Final Answer

(E_1)=span*{[[1],[0],[0],[0]],[[0],[1],[0],[0]]},(E_0)=span*{[5],[0],[1],[0]}

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