Show that r is repeated root for characteristic equation iff

In summary, the conversation discusses the relationship between the kernel of a linear operator A - rI and its power (A - rI)^2, where r is a multiple root for the minimal polynomial u(x). It is stated that the kernel of (A - rI)^2 contains the eigenvectors associated with r, but not the kernel of (A - rI). The concept of a generalized eigenvector is introduced and it is suggested to use the fact that (x-r)^2 divides the minimal polynomial to find a vector v2 that satisfies (A - rI)v2 = v.
  • #1
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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)?
 
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  • #2
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 [itex]\in[/itex] ker( (A - rI)2). But v2 [itex]\notin[/itex] 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.
 

FAQ: Show that r is repeated root for characteristic equation iff

1. What is a repeated root in a characteristic equation?

A repeated root in a characteristic equation refers to a situation where one of the roots of the equation is repeated multiple times. This means that the equation has a root with a multiplicity greater than one.

2. How can you show that r is a repeated root for a characteristic equation?

In order to show that r is a repeated root for a characteristic equation, you can use the factor theorem. This theorem states that if a polynomial function has a root at r, then (x-r) is a factor of the polynomial. Therefore, if (x-r) is a factor of the characteristic equation, then r is a root with multiplicity greater than one.

3. What does it mean for a root to have a multiplicity greater than one?

Having a root with a multiplicity greater than one means that the root appears multiple times in the equation. For example, if the root is 3 and it has a multiplicity of 2, then the equation will have (x-3)^2 as a factor, indicating that 3 appears twice as a root.

4. What is the significance of identifying a repeated root in a characteristic equation?

Identifying a repeated root in a characteristic equation can help in determining the stability of a system described by the equation. If the root has a multiplicity greater than one, it can affect the behavior of the system and lead to different outcomes. It also helps in finding the general solution to the equation.

5. Can a characteristic equation have more than one repeated root?

Yes, it is possible for a characteristic equation to have more than one repeated root. This means that there can be multiple values of r that make (x-r) a factor of the equation. In this case, the equation will have (x-r)^k as a factor, where k is the multiplicity of the root r.

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