Sparsity of a matrix

Definition

Suppose $m,n$ are positive integers and $A$ is a $m\times n$ matrix. The sparsity or sparseness of $A$ is defined as the fraction of entries of $A$ that have value 0. Explicitly, it is the ratio:

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

The sparsity of the matrix can also be defined as 1 minus its density. Formulas about the relation with matrix operations are more neatly expressed in terms of density than sparsity.

See also

• Sparse matrix family, a family of matrices whose sparsity approaches 1 in the limit.