Homework Help: Show that r is repeated root for characteristic equation iff

1. Feb 6, 2014

1. The problem statement, all variables and given/known data
A:B→B a linear operator

Show r is multiple root for minimal polynomial u(x) iff

>$$\{0\}\subset \ker(A - rI) \subset \ker(A - rI)^2$$

note: it is proper subset

2. Relevant equations

3. The attempt at a solution
1. The problem statement, all variables and given/known data

My thought:

I know ker(A−rI) is basically {{0} and {eigenvectors associated with r}}.

what is ker((A−rI)^2) with respect to above and/or r? How is eigenvector of (A−rI)^2 related to that of (L−rI)?

Last edited: Feb 6, 2014
2. Feb 7, 2014

kduna

Have you ever heard of a generalized eigenvector?

http://en.wikipedia.org/wiki/Generalized_eigenvector

Suppose v is an eigenvector corresponding corresponding to r. If you could find a vector v2 such that (A - rI) v2 = v. Then:

(A - rI)2 v2 = (A - rI) v = 0​

So v2 $\in$ ker( (A - rI)2). But v2 $\notin$ ker(A - rI).

See if you can use the fact that (x-r)2 divides the minimal polynomial to show that such a v2 exists.