Density of a matrix

Definition

Suppose $m,n$ are positive integers and $A$ is a $m\times n$ matrix. The density of $A$ is defined as the fraction of entries of $A$ that have nonzero value. Explicitly, it is the ratio:

${\frac {|\{(i,j)\in \{1,2,\dots ,m\}\times \{1,2,\dots ,n\}:a_{ij}\neq 0\}|}{mn}}$

The density of the matrix can also be defined as 1 minus its sparsity.

The density of a matrix can be any rational number in $[0,1]$. For a $m\times n$ matrix, the rational number must be expressible as an integer divided by $mn$.

Relation with other matrix measures

Measure	Numerical relationship with density for a $m\times n$ matrix ($n\times n$ if it is required to be square)
sparsity of a matrix	The density and sparsity add up to 1.
rank of a matrix	The density is at least equal to rank$/(mn)$.
bandwidth of a matrix (for a square matrix)	The bandwidth is at least equal to $n$ times the density (this can be improved somewhat by a constant order of magnitude of about 2).

Matrix operations

Matrix operation	Lower bound on density of result	Case where this occurs	Upper bound on density of result	Case where this occurs
Addition of two $m\times n$ matrices $A$ and $B$ with densities $d_{A}$ and $d_{B}$ respectively	$|d_{A}-d_{B}|$	For all positions where both entries are nonzero, they are negatives of one another.	$d_{A}+d_{B}$ Can be refined to $\min\{d_{A}+d_{B},1\}$	$d_{A}+d_{B}$ occurs when the set of positions with nonzero entries for $A$ is disjoint from the corresponding set for $B$.
Multiplication of a $m\times n$ matrix $A$ and a $n\times p$ matrix $B$	$d_{A}d_{B}</math.if<math>n=1$ 0 if $n>1$	The $n=1$ case boils down to a Hadamard product computation. For $n>1$, we can arrange for a zero product regardless of density by choosing the rows of $A$ and the columns of $B$ to be orthogonal.	$n^{2}d_{A}d_{B}$ Can be refined to $\min\{n^{2}d_{A}d_{B},1\}$	This occurs if there is exactly one column with nonzero entries in $A$, and exactly one row with nonzero entries in $B$, and the index number of the column and row coincide. The product $AB$ in this case coincides with the Hadamard product of the nonzero column and the nonzero row.
Matrix transpose of a $m\times n$ matrix $A$, giving a $n\times m$ matrix $A^{T}$	$d_{A}$	bound always attained precisely	$d_{A}$	bound always attained precisely
Inverse matrix of a $n\times n$ matrix $A$	? (at least $1/n$)	?	?	?

|}