Show that r is repeated root for characteristic equation iff

[catsarebad]

Homework Statement

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

Homework Equations

The Attempt at a Solution

Homework Statement

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)?

 

[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.