In mathematics, a distance matrix is a matrix (two-dimensional array) containing the distances, taken pairwise, of a set of points. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and In Mathematics, a matrix (plural matrices) is a rectangular table of elements (or entries) which may be Numbers or more generally In Computer science an array is a Data structure consisting of a group of elements that are accessed by indexing. Distance is a numerical description of how far apart objects are It is therefore a symmetric N×N matrix containing non-negative reals as elements, given N points in Euclidean space. Symmetry generally conveys two primary meanings The first is an imprecise sense of harmonious or aesthetically-pleasing proportionality and balance such that it reflects beauty or The number of pairs of points N×(N-1)/2 is the number of independent elements in the distance matrix.

Distance matrices are related to 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 an distance matrix is smaller if two elements are closer, while "close" (connected) vertices yield larger entries in an adjacency matrix. In Mathematics and Computer science, the adjacency matrix of a finite directed or undirected graph G on n vertices is the

For example, suppose these data are to be analyzed. Where pixel euclidean distance is the distance metric. In Digital imaging, a pixel ( pict ure el ement is the smallest piece of information in an image In Mathematics, the Euclidean distance or Euclidean metric is the "ordinary" Distance between two points that one would measure with a ruler In Mathematics, a metric or distance function is a function which defines a Distance between elements of a set.

The distance matrix would be:

	a	b	c	d	e	f
a	0	184	222	177	216	231
b	184	0	45	123	128	200
c	222	45	0	129	121	203
d	177	123	129	0	46	83
e	216	128	121	46	0	83
f	231	200	203	83	83	0

These data can then be viewed in graphic form as a heat map. In this image, black denotes a distance of 0 and white is maximal distance.

In bioinformatics, distance matrices are used to represent protein structures in a coordinate-independent manner, as well as the pairwise distances between two sequences in sequence space. Bioinformatics is the application of information technology to the field of molecular biology Proteins are large Organic compounds made of Amino acids arranged in a linear chain and joined together by Peptide bonds between the Carboxyl They are used in structural and sequential alignment, and for the determination of protein structures from NMR or X-ray crystallography. Structural alignment is a form of Sequence alignment based on comparison of shape In Bioinformatics, a sequence alignment is a way of arranging the Primary sequences of DNA, RNA, or Protein to identify regions of X-radiation (composed of X-rays) is a form of Electromagnetic radiation. Crystallography is the experimental science of determining the arrangement of Atoms in Solids In older usage it is the scientific study of Crystals The

Sometimes it is more convenient to express data as a similarity matrix. A similarity matrix is a matrix of scores which express the similarity between two data points

See also

• Data clustering
• Computer Vision

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