Calculate the eigenvectors of a specific matrix

[Dassinia]

Hello,
I'm really having a problem to calculate the eigenvectors of a specific matrix, I'm used to do this but i don't know why I'm stuck at this one

Homework Statement

A=
2 1 0 1
0 3 -1 0
0 1 1 0
0 -1 0 3

λ₁=2 multiplicity 3
λ₂=3 multiplicity 1

A-2I=
0 1 0 1
0 1 -1 0
0 1 -1 0
0 -1 0 1

The eigenvectors are given
[-1 -1 -1 -1]
[0 1 2 0]
[1 0 0 0]

If I solve
(A-2I)xi=0
I have
x1=0
x2+x4=0
x2-x3=0
-x2+x4=0


I have to use this for differential equations, and my linear algebra course is far behind, I don't remember what I'm supposed to do when we get the first eigenvector =0, because when the multiplicity of the eigenvalue is > 0 we use the first one to find the following eigenvectors

Thanks!

 

[Dick]

Hello,
I'm really having a problem to calculate the eigenvectors of a specific matrix, I'm used to do this but i don't know why I'm stuck at this one

Homework Statement

A=
2 1 0 1
0 3 -1 0
0 1 1 0
0 -1 0 3

λ₁=2 multiplicity 3
λ₂=3 multiplicity 1

A-2I=
0 1 0 1
0 1 -1 0
0 1 -1 0
0 -1 0 1

The eigenvectors are given
[-1 -1 -1 -1]
[0 1 2 0]
[1 0 0 0]

If I solve
(A-2I)xi=0
I have
x1=0
x2+x4=0
x2-x3=0
-x2+x4=0


I have to use this for differential equations, and my linear algebra course is far behind, I don't remember what I'm supposed to do when we get the first eigenvector =0, because when the multiplicity of the eigenvalue is > 0 we use the first one to find the following eigenvectors

Thanks!

There is a nonzero eigenvector corresponding to eigenvalue 2. Rethink your conclusion that solving that matrix gives you x1=0.