  Position operator $\hat{r}$ appears as $i{\partial_p}$ in wave mechanics,
while its matrix form is well known diverging in diagonals, causing serious
difficulties in basis transformation, observable yielding, etc. We aim to find
a convergent $r$-matrix (CRM) to improve the existing divergent $r$-matrix
(DRM), and investigate its influence at both the conceptual and the application
levels. Unlike the spin matrix, which affords a Lie algebra representation as
the solution of $[s_i,s_j]={\epsilon}_{i,j,k}s_k$, the $r$-matrix cannot be a
solution for $[\hat{r},p]=i\hbar$, namely Weyl algebra. Indeed: matrix
representations of Weyl algebras prove not existing; thus, neither CRM nor DRM
would afford a representation. Instead, the CRM should be viewed as a procedure
of encoding $\hat{r}$ using matrices of arbitrary finite dimensions. Deriving
CRM recognizes that the limited understanding about Weyl algebra has led to the
divergence. A key modification is increasing the 1-st Weyl algebra (the
familiar substitution $\hat{r}{\rightarrow}i{\partial_p}$) to the $N$-th Weyl
algebra. Resolving the divergence makes $r$-matrix rigorously defined, and we
are able to show $r$-matrix is distinct from a spin matrix in terms of its
defining principles, transformation behavior, and the observable it yields. At
the conceptual level, the CRM fills the logical gap between the $r$-matrix and
the Berry connection; and helps to show that Bloch space $\mathcal{H}_B$ is
incomplete for $\hat{r}$. At the application level, we focus on transport, and
discover that the Hermitian matrix is not identical with the associative
Hermitian operator, i.e.,
$r_{m,n}=r_{n,m}^*{\nLeftrightarrow}\hat{r}=\hat{r}^{\dagger}$. We also discuss
how such a non-representation CRM can contribute to building a unified
transport theory.
