Proving a property of eigenvalues and their eigenvectors.

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Homework Help Overview

The discussion revolves around proving a property of eigenvalues, specifically that if λ is an eigenvalue of matrix A, then λ + k is an eigenvalue of the matrix A + kI, where I is the identity matrix. Participants are exploring the implications of this property in the context of linear algebra.

Discussion Character

  • Conceptual clarification, Mathematical reasoning

Approaches and Questions Raised

  • Participants are attempting to manipulate the eigenvalue equation and express the relationship between the eigenvalues of A and A + kI. There is a focus on identifying errors in reasoning and clarifying the assumptions made during the proof process.

Discussion Status

The discussion is active, with participants providing feedback on each other's attempts. One participant has pointed out an error in the reasoning of another, leading to a realization about the assumptions made regarding eigenvalues. There is an ongoing exploration of the correct relationships and definitions involved.

Contextual Notes

Participants are working under the constraints of homework guidelines, which may limit the amount of direct assistance they can provide to each other. There is a recognition of the need to clarify definitions and properties of eigenvalues in the context of the problem.

pondzo
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Homework Statement



I am asked to prove that if λ is an eigenvalue of A then λ + k is an eigenvalue of
A + kI.

The Attempt at a Solution



## A\vec{v}=\lambda\vec{v} ##

## (A+kI)\vec{v}=\lambda\vec{v} ##
## A\vec{v}+k\vec{v} = \lambda\vec{v} ## → ## A\vec{v} = \lambda\vec{v} - k\vec{v} ##
## A\vec{v} = (\lambda-k)\vec{v} ## however, this is not λ+k
I must be missing something pretty obvious, but i can't see what it is...
 
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Your second step is the error. You have assumed in that step that lambda is an eigenvalue of A+kI in that step.
 
Ahh thank you! I didn't even realize that, I am so use to denoting the eigenvalues lambda..
 
pondzo said:

Homework Statement



I am asked to prove that if λ is an eigenvalue of A then λ + k is an eigenvalue of
A + kI.

The Attempt at a Solution



## A\vec{v}=\lambda\vec{v} ##

## (A+kI)\vec{v}=\lambda\vec{v} ##
## A\vec{v}+k\vec{v} = \lambda\vec{v} ## → ## A\vec{v} = \lambda\vec{v} - k\vec{v} ##
## A\vec{v} = (\lambda-k)\vec{v} ## however, this is not λ+k
I must be missing something pretty obvious, but i can't see what it is...

If ##\lambda + k## is an eigenvalue of ##A + kI##, then:

##Av = \lambda v##
##Av + (kI)v = \lambda v + (kI)v##
##(A + kI)v = (\lambda + kI)v##
 
Last edited:

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