TITLE: Decomposing a matrix into a product of sparse matrices
QUESTION [5 upvotes]: How to study the decomposition of a square matrix into a product of sparse matrices? 
There are no restrictions on the number of matrices in the product, but the fewer the better.

REPLY [10 votes]: Given an invertible $n \times n$ matrix $\mathrm A$, we perform Gaussian elimination until we obtain a (nonsingular) diagonal matrix. In other words, we left-multiply $\mathrm A$ by permutation matrices (whose inverses are their transposes) and by elementary matrices of the form
$$\mathrm E_{ij} := \mathrm I_n + \beta_{ij} \, \mathrm e_i \mathrm e_j^{\top}$$
whose inverse is simply
$$\mathrm E_{ij}^{-1} = \mathrm I_n - \beta_{ij} \, \mathrm e_i \mathrm e_j^{\top}$$
until we obtain a (nonsingular) diagonal matrix $\mathrm D$, i.e.,
$$\mathrm E_{i_m j_m} \mathrm P_m \cdots \mathrm E_{i_2 j_2} \mathrm P_2 \mathrm E_{i_1 j_1} \mathrm P_1 \mathrm A = \mathrm D$$
where $m$, the number of $\mathrm E_{i j}$ elementary matrices to the left of $\mathrm A$, is at most $n (n-1)$. Thus, one decomposition of $\mathrm A$ into a product of sparse matrices is
$$\boxed{\mathrm A = \mathrm P_1^{\top} \mathrm E_{i_1 j_1}^{-1} \mathrm P_2^{\top} \mathrm E_{i_2 j_2}^{-1} \cdots \mathrm P_m^{\top} \mathrm E_{i_m j_m}^{-1} \mathrm D}$$
where the $\mathrm E_{i j}^{-1}$ matrices are sparse and easy to compute. The fraction of nonzero entries in these matrices is $n^{-1} + n^{-2} \approx n^{-1}$ for the $\mathrm E_{i j}^{-1}$ matrices and $n^{-1}$ for matrix $\mathrm D$ and for each permutation matrix. Hence, the larger $n$, the sparser the matrices.
Multiplying the sparse elementary matrices pertaining to the same column (e.g., $\mathrm E_{21}, \mathrm E_{31}, \dots, \mathrm E_{n1}$ for the first column), we obtain less sparse Frobenius matrices. To buy a smaller number of factors in the decomposition, we pay with sparsity.
Though the $\mathrm E_{i j}$ and $\mathrm E_{i j}^{-1}$ matrices have $n+1$ nonzero entries, only one of them, the $(i,j)$-th entry, is in general different from $1$. Therefore, we only need one real number, $\beta_{ij}$, and one pair of positive integers, $(i,j)$, to fully specify the elementary matrix $\mathrm E_{i j}$ (or its inverse). Taking into account also the $n$ nonzero entries on the main diagonal of the diagonal matrix $\mathrm D$, much of the data needed to build the factors in the decomposition can be stored in the following $n \times n$ matrix
$$\begin{bmatrix} d_1 & \beta_{12} & \dots & \beta_{1n}\\ \beta_{21} & d_2 & \dots & \beta_{2n}\\ \vdots & \vdots & \ddots & \vdots\\ \beta_{n1} & \beta_{n2} & \dots & d_n\end{bmatrix}$$
Each permutation matrix can be specified using $n$ positive integers. Thus, at most $n^2$ reals and $n^2$ integers are needed to specify the decomposition.

Example
Consider the (invertible) $3 \times 3$ magic square
$$\mathrm A := \begin{bmatrix} 2 & 9 & 4\\7 & 5 & 3\\6 & 1 & 8\end{bmatrix}$$
Using SymPy,
>>> from sympy import *
>>> A = Matrix([[2,9,4],[7,5,3],[6,1,8]])

Eliminate two entries below the $(1,1)$-th entry:
>>> E21 = eye(3) + Matrix([[0,0,0],[-7.0/2.0,0,0],[0,0,0]])
>>> E31 = eye(3) + Matrix([[0,0,0],[0,0,0],[-3.0,0,0]]) 
>>> E31 * E21 * A
Matrix([
[2,     9,     4],
[0, -26.5, -11.0],
[0, -26.0,  -4.0]])

Eliminate the entry below the $(2,2)$-th entry:
>>> E32 = eye(3) + Matrix([[0,0,0],[0,0,0],[0,-26.0/26.5,0]])
>>> E32 * E31 * E21 * A
Matrix([
[2,                    9,                4],
[0,                -26.5,            -11.0],
[0, 1.33226762955019e-15, 6.79245283018868]])

Numerical noise is now appearing. Eliminate two entries above the $(3,3)$-th entry
>>> E23 = eye(3) + Matrix([[0,0,0],[0,0,11.0/6.79245],[0,0,0]])
>>> E13 = eye(3) + Matrix([[0,0,-4.0/6.79245],[0,0,0],[0,0,0]])
>>> E13 * E23 * E32 * E31 * E21 * A
Matrix([
[2.0,                  9.0, -1.66666736145515e-6],
[  0,                -26.5,  4.58333524377963e-6],
[  0, 1.33226762955019e-15,     6.79245283018868]])

Eliminate the entry above the $(2,2)$-th entry:
>>> E12 = eye(3) + Matrix([[0,9.0/26.5,0],[0,0,0],[0,0,0]])
>>> E12 * E13 * E23 * E32 * E31 * E21 * A
Matrix([
[2.0, 1.99840144432528e-15, -1.10062938096789e-7],
[  0,                -26.5,  4.58333524377963e-6],
[  0, 1.33226762955019e-15,     6.79245283018868]])

We extract the diagonal:
>>> D = diag(2.0,-26.5,6.79245)
>>> D
Matrix([
[2.0,     0,       0],
[  0, -26.5,       0],
[  0,     0, 6.79245]])

Finally, we reconstruct the magic square:
>>> (E21**-1) * (E31**-1) * (E32**-1) * (E23**-1) * (E13**-1) * (E12**-1) * D
Matrix([
[2.0, 9.0,              4.0],
[7.0, 5.0,              3.0],
[6.0, 1.0, 7.99999716981132]])

The reconstruction is not exact due to numerical noise, but it is close enough. Thus, magic square $\mathrm A$ can be approximately factored as follows
$$\underbrace{\begin{bmatrix} 1 & 0 & 0\\ 3.5 & 1 & 0\\3 & 0 & 1\end{bmatrix}}_{= \mathrm E_{21}^{-1} \mathrm E_{31}^{-1}} \, \begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0\\0 & 0.98 & 1\end{bmatrix} \, \underbrace{\begin{bmatrix} 1 & 0 & 0.59\\0 & 1 & -1.62\\0 & 0 & 1\end{bmatrix}}_{\approx \mathrm E_{23}^{-1} \mathrm E_{13}^{-1}} \, \begin{bmatrix} 1 & -0.34 & 0\\0 & 1 & 0\\0 & 0 & 1\end{bmatrix} \, \underbrace{\begin{bmatrix} 2 & 0 & 0\\ 0 & -26.5 & 0\\0 & 0 & 6.79\end{bmatrix}}_{\approx \mathrm D}$$
where the 1st and 3rd of the five matrix factors are Frobenius matrices. In this particular example, no permutation matrices (other than $\mathrm I_n$, that is) were needed.