Let $\mathbb{F}$ be a non-Archimedean local field. Let $\{T\_a\}\_{a=1}^\infty$ be a sequence of linear operators $\mathbb{F}^n\to\mathbb{F}^n$ of rank $n$. After a choice of subsequence, is it possible to construct sequences of vectors $\{e^i\_a\}\subset \mathbb{F}^n$, $i=1,\dots, n,$ such that the following properties are satisfied:


(1) for any $a$ the vectors $e^1\_a,\dots, e^n\_a$ form a basis of $\mathbb{F}^n$ and when $a\to \infty$ they converge to another basis of $\mathbb{F}^n$;


(2) the vectors $T\_a(e^1\_a),\dots, T\_a(e^n\_a)$ also form a basis of $\mathbb{F}^n$ and when $a\to \infty$ appropriate multiples of these vectors converge to another basis of $\mathbb{F}^n$?


Remark. In the Archimedean case the answer is positive. Indeed fix a Euclidean (resp. Hermitian) metric. $T\_a$ admits an orthonormal basis $e^1\_a,\dots, e^n\_a$ such that $T\_a(e^1\_a),\dots, T\_a(e^n\_a)$ are pairwise orthogonal vectors. After a choice of subsequence all the required properties will be satisfied.



A question on linear algebra over non-Archimedean local field
$\def\FF{\mathbb{F}}\def\GL{\text{GL}}$This is basically the same thing YCor sketches in his comment: Let $R$ be the ring of integers of $\FF$. Let $K$ be the group of invertible matrices with entries in $R$ whose inverses are also in $R$, and let $T$ be the group of diagonal matrices.


The ring $R$ is a dvr so, by the [Smith normal form theorem](https://en.wikipedia.org/wiki/Smith_normal_form), each of your matrices $T\_a$ can be factored as $U\_a D\_a V\_a$ with $D\_a \in T$ and $U\_a$, $V\_a \in K$.


The group $K$ is compact in the non-archimedean topology. Proof: $K$ can be written as $\{ (g,h) : gh = \text{Id}\_n,\ g,h \in \text{Mat}\_{n \times n}(R) \}$. This is a closed subspace of $\text{Mat}\_{n \times n}(R)^2$, and $\text{Mat}\_{n \times n}(R)^2 \cong R^{2n^2}$ is obviously compact. So we can extract a subsequence where the $U\_a$ approach a limit $U$ and the $V\_a$ approach a limit $V$.


Then we take $(e\_a^1, e\_a^2, \ldots, e\_a^n)$ to be the columns of $V\_a^{-1}$, just as in the archimedean case.


I learned the slogan "Smith normal form is non-archimedean singular value decomposition" from Kiran Kedlaya.


