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In [[mathematics]],، [[computer science]] and [[graph theory]],، a '''distance matrix''' is a [[matrix (mathematics)|matrix]] (two-dimensional array) containing the [[distance]]s,s، taken pairwise,pairwise، of a set of points. This matrix will have a size of ''N''××''N'' where ''N'' is the number of points,points، nodes or vertices (often in a graph).

== Comparison with related matrices ==

=== Comparison with Adjacency matrix ===

Distance matrices are related to [[Adjacency matrix|adjacency matrices]],، with the differences that (a) the latter only provides the information which vertices are connected but does not tell about ''costs'' or ''distances'' between the vertices and (b) an entry of a distance matrix is smaller if two elements are closer,closer، while "close" (connected) vertices yield larger entries in an adjacency matrix.

=== Comparison with Euclidean distance matrix ===

Unlike a [[Euclidean distance matrix]],، the matrix does not need to be [[Symmetric matrix|symmetric]] -- that is,is، the values ''x_i,ji،j'' do not necessarily equal ''x_j,ij،i''. Similarly,Similarly، the matrix values are not restricted to non-negative [[Real number|reals]] (as they would be in the Euclidean distance matrix) but rather can have negative values,values، zeros or [[imaginary number]]s depending on the cost metric and specific use. Although it is often the case,case، distance matrices are not restricted to being [[hollow matrix|hollow]] -- that is,is، they can have non-zero entries on the main diagonal.

== Examples and uses ==

For example,example، suppose these data are to be analyzed,analyzed، where [[pixel]] [[euclidean distance]] is the [[Metric (mathematics)|distance metric]].

[[Imageپرونده:Clusters.svg|frame|none|Raw data]]

These data can then be viewed in graphic form as a [[heat map]]. In this image,image، black denotes a distance of 0۰ and white is maximal distance.

[[Imageپرونده:Distance matrix.PNG|frame|none|Graphical View]]

In [[bioinformatics]],، distance matrices are used to represent [[protein]] structures in a coordinate-independent manner,manner، as well as the pairwise distances between two sequences in sequence space. They are used in [[structural alignment|structural]] and [[sequence alignment|sequential]] alignment,alignment، and for the determination of protein structures from [[Nuclear magnetic resonance|NMR]] or [[X-ray crystallography]].

== See also ==