Sparsity of a matrix: Difference between revisions
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Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''sparsity''' or '''sparseness''' of <math>A</math> is defined as the fraction of entries of <math>A</math> that have value 0. Explicitly, it is the ratio: | Suppose <math>m,n</math> are positive integers and <math>A</math> is a <math>m \times n</math> matrix. The '''sparsity''' or '''sparseness''' of <math>A</math> is defined as the fraction of entries of <math>A</math> that have value 0. Explicitly, it is the ratio: | ||
<math>\frac{ | <math>\frac{|\{ (i,j) \in \{ 1,2,\dots,m\} \times \{1,2,\dots,n \} : a_{ij} = 0 \}|}{mn}</math> | ||
The sparsity of the matrix can also be defined as 1 minus its [[density of a matrix|density]]. Formulas about the relation with matrix operations are more neatly expressed in terms of density than sparsity. | The sparsity of the matrix can also be defined as 1 minus its [[density of a matrix|density]]. Formulas about the relation with matrix operations are more neatly expressed in terms of density than sparsity. | ||
Latest revision as of 15:53, 1 May 2014
Definition
Suppose are positive integers and is a matrix. The sparsity or sparseness of is defined as the fraction of entries of that have value 0. Explicitly, it is the ratio:
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.