Proving a property of eigenvalues and their eigenvectors.

1. May 24, 2014

pondzo

1. The problem statement, all variables and given/known data

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

3. 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...

2. May 24, 2014

Matterwave

Your second step is the error. You have assumed in that step that lambda is an eigenvalue of A+kI in that step.

3. May 24, 2014

pondzo

Ahh thank you! I didn't even realise that, im so use to denoting the eigenvalues lambda..

4. May 24, 2014

Zondrina

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: May 24, 2014