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.