TITLE: Does $Ax=x$ imply $A^* x=x$, if $A^*$ is the conjugate transpose of $A$?
QUESTION [6 upvotes]: I have a fairly simple question. If $A$ is a matrix and $A^*$ denotes its conjugate transpose, is it true that if $Ax = x$, then $A^*x = x$?
The matrix $A^*$ will certainly have $1$ as an eigenvalue, but will it be with the same eigenvector? And if not, what is the relation between the eigenvector of $A$ and the one of $A^*$?

REPLY [4 votes]: The easiest example would be to consider the rank one matrix
$$A = xy^\top$$
Then $$Ax = (xy^\top) x= x(y^\top x) = x\lambda = \lambda x$$
and
$$ A^* x = (\bar{y} \bar{x}^\top) x =\bar{y} (\bar{x}^\top x )= \bar{y} k = k\bar{y}$$