Decomposition of a Matrix by Sparse Matrices Let $\mathbb{F}$ is a field. Consider an $n \times n$ matrix $\bf A$ over  $\mathbb{F}$.  
$\bf A$ is called sparse matrix over $\mathbb{F}$ iff the number of non-zero entities of $\bf A$  be at most $2n$. 
My question:
Consider a non-zero $n \times n$ matrix $\bf M$ over $\mathbb{F}$. Is there a method or an algorithm such that $\bf M$ can be decomposed as follows: 
$$
{\bf M}=\prod_{i=1}^n\, {\bf A}_i=A_1\,A_2\, \cdots \,A_n\, .
$$
where ${\bf A}_i$'s are sparse $n \times n$ matrices over $\mathbb{F}$. (I need binary finite field or $\mathbb{F}_{2^q}$)
For simplicity, we can assume that ${\bf A}_i$'s have the same sparsity pattern. 
I appreciate to address me  paper or book about this subject.  
Thanks for any suggestions. 

Over any field, you can do the job with $2n-2$ "sparse" (according to your definition) matrices. Behind lies the decomposition LU.
Take $A_1,\cdots A_{n-1}$ lower triangular with $2$ bands and diagonal vector $[1,\cdots,1]$. Take $A_n,\cdots A_{2n-2}$ upper triangular with $2$ bands. 
