Vipul at 15:53, 1 May 2014

2014-05-01T15:53:56Z

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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{# \{ (i,j) \in \{ 1,2,\dots,m\} \times \{1,2,\dots,n \} : a_{ij} = 0 \}}{mn}</math>		<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.

Vipul: Created page with "==Definition== Suppose $m,n$ are positive integers and $A$ is a $m \times n$ matrix. The '''sparsity''' or '''sparseness''' of $A$ ..."

2014-05-01T15:53:20Z

Created page with "==Definition== 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>..."

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==Definition==

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{# \{ (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.

==See also==

* [[Sparse matrix family]], a family of matrices whose sparsity approaches 1 in the limit.

Sparsity of a matrix - Revision history

Vipul at 15:53, 1 May 2014

Vipul: Created page with "==Definition== Suppose m,n are positive integers and A is a m \times n matrix. The '''sparsity''' or '''sparseness''' of A..."

Vipul: Created page with "==Definition== Suppose $m,n$ are positive integers and $A$ is a $m \times n$ matrix. The '''sparsity''' or '''sparseness''' of $A$ ..."