A $n \times n$ matrix $\bf A$ over the field $\mathbb{F}$ is sparse iff the number of non-zero entries of $\bf A$ is at most $2n$.
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
$$ {\bf M}=\prod_{i=1}^n\, {\bf A}_i = {\bf A}_1 \, {\bf A}_2 \, \cdots \, {\bf A}_n$$
where ${\bf A}_i$'s are sparse $n \times n$ matrices over $\mathbb{F}$? I need binary finite field $\mathbb{F}_{2^q}$. For simplicity, we can assume that the matrices ${\bf A}_i$ have the same sparsity pattern.
I appreciate references to papers or books about this subject.
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.