TITLE: Ambiguity of the Eigenvalue
QUESTION [5 upvotes]: For matrix $A$ let $\lambda$ be eigenvalue corresponding to eigenvector $v$.
$$Av = \lambda v = \frac{c}{1} \lambda \frac{1}{c} v = \mu \frac{1}{c} v,$$
where $c$ is some real number. It is clear that if $c \neq 1$, then $\mu \neq \lambda$. This would imply that the eigenvalue can be any real number and it is only the direction of $v$ that matters? Without normalization we could not talk about the greatest eigenvalue?
EDIT:
$$\mu \frac{1}{c} v = \mu u = Av = cAu$$
thus $Au \neq \mu u$, if $c \neq 1$.

REPLY [3 votes]: That is not true because your "new" eigenvalue is $\mu \frac{1}{c}$ (not $\mu$) which keep beeing $\lambda$.
$\mu \frac{1}{c}=\lambda$

REPLY [2 votes]: You wrote $Av = \mu \frac 1 c v$, hence by definition of an eigenvector
$$A \frac 1c v = \frac 1c\mu\frac 1c v ,$$ so the eigenvalue is $\frac \mu c=\lambda$.