Let $F$ denote a finite field and $A$ a square matrix with coefficients in $F$. The set of all matrices $B$ such that $BA=AB$ is called the centraliser of $A$.
Now consider the set $C(a,A)$ of all matrices $B$ satisfying $AB=aBA$ where $a$ is in $F$. We call this set the twisted centraliser of $A$. It is an $F$-vector space. Are there bounds in the linear algebra literature on its dimension?