I'm facing the following problem:
Let $A$ be an $m\times n$ matrix over a commutative ring $R$ (to begin with, a finite field would be sufficient too) and want to compute a decomposition in terms of elementary tensors (or in other words rank 1 matrices). That is, $$A=\sum_{i=1}^{r} a_i\otimes b_i$$ with $a_i\in R^m$, $b_i\in R^n$ and $r$ is the rank of the matrix.
All literature I found was about matrices over $\mathbb{R}$, $\mathbb{C}$, or about singular value decomposition and so on, but nothing that is applicable to finite fields or rings. I'm looking for algorithms and literature about this problem. Does anyone have an idea?
Cheers.
I think the following construction covers many a ring $R$, at least all the fields and I think also PIDs.
Assume that the row space of $A$ is, or is contained in, a free $R$-module $V\subseteq R^n$ of rank $r$. Let $a_1,a_2,\ldots,a_r$ be a basis of $V$. So if we consider the row $v_j, j=1,2,\ldots,m$, then there exist coefficients $b_{ij}\in R$ such that $$ v_j=\sum_ib_{ij}a_i. $$ Let then $b_i$ be the column vector $(b_{ij})_{1\le j\le m}$. The elementary tensor $a_i\otimes b_i$ is then a matrix with the $j$th row equal to $b_{ij}a_i$. Therefore $$A=\sum_i a_i\otimes b_i.$$