# Math Help - eigenvector of A+I

1. ## eigenvector of A+I

I need to either prove or give a counterexample of the following: If x is an eigenvector of A, then x is also an eigenvector of A+I. I tried several different matrices, and this proposition does seem to be true. However, I have no idea how to prove it.

Help!

2. Suppose x is an eigenvector of $A$, ie. $\exists \lambda \in \mathbb{R} \ : \ Ax = \lambda x \Rightarrow (A + I)x = Ax + x = ...$
And finish.

3. If $x$ is an eigen vector of a matrix $A$ corresponding to an eigen value $\lambda$

$\therefore Ax=\lambda x$

$\therefore Ax+x=\lambda x+x$

$\therefore (A+I)x=(\lambda +1)x$

$\therefore (A+I)x=\lambda_1 x$, where $\lambda_1=\lambda+1$

$\therefore x$ is an eigen vector of a matrix $A+I$ corresponding to an eigen value $\lambda_1=\lambda+1$.

Proved.