How can find this matrix?

Question

Let $U$ be an $k\times n$ matrix and $G$ an $n \times n$ matrix over $\mathbb F_q$. We know that valuation of $UG$,$UG^2$,...,$UG^{k-1}$. ($k$ is the order of $G$) If we dont know components of $U$ and $G$, can I find $U$ or $G$?

Answer 1

Note that $I+G+G^2+\ldots+G^{k-1} = (I-G^k)(I-G)^{-1}$ (geometric series, if $I-G$ is invertible)

And $I = G^k$ implies that the RHS is 0.

So we have $I = -(G+G^2+\ldots+G^{k-1})$ and therefore

$$U= -(UG+UG^2+\ldots+UG^{k-1}).$$