Computing eigenvalues and eigenvectors

[jinksys]

Find the characteristic polynomial, eigenvectors, and eigenvalues of the matrix.

[
2 -2 3
0 3 -2
0 -1 2
]

My characteristic poly is x^3 - 7x^2 + 14x - 8

The roots are 1, 2, and 4.

When solving the system, (2I - A)x = 0 I get:

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

Can someone tell me how this relates to a solution?

It looks like x2 and x3 would be zero, but what about x1?

 

[hgfalling]

This means that (t 0 0) is an eigenvector for any t. Since constant multiples of eigenvectors are also eigenvectors, typically we would say that (1 0 0) is the eigenvector (associated with the eigenvalue 2).

 

[jinksys]

Thanks, that makes sense.