Let $P\in R^{n\times n}$ be an orthogonal matrix. I want to ask whether or not there exists some vector $x\in R^n$ containing no zero entries such that $Px$ also contains no zero entries.



A question on orthogonal matrix
Instead of Zariski topology consider usual topology. Every open ball will contain a vector with all entries non-zero. (I leave this to you to verify).
As $P$ is invertible it is a homeomorphism. Image of open ball will contain open ball. QED


