Distance matrix

In mathematics, computer science and especially graph theory, a distance matrix is a square matrix (two-dimensional array) containing the distances, taken pairwise, between the elements of a set. Depending upon the application involved, the "distance" being used to define this matrix may or may not be a metric. If there are elements, this matrix will have size . In graph-theoretic applications the elements are more often referred to as points, nodes or vertices.
Non-metric distance matrices.
In general, a distance matrix is a weighted adjacency matrix of some graph. In a network, a directed graph with weights assigned to the arcs, the distance between two nodes of the network can be defined as the minimum of the sums of the weights on the shortest paths joining the two nodes. This distance function, while well defined, is not a metric. There need be no restrictions on the weights other than the need to be able to combine and compare them, so negative weights are used in some applications. Since paths are directed, symmetry can not be guaranteed, and if cycles exist the distance matrix may not be hollow.
An algebraic formulation of the above can be obtained by using the min-plus algebra. Matrix multiplication in this system is defined as follows: Given two formula_1 matrices formula_2 and formula_3, their distance product formula_4 is defined as an formula_1 matrix such that formula_6. Note that the off-diagonal elements that are not connected directly will need to be set to infinity or a suitable large value for the min-plus operations to work correctly. A zero in these locations will be incorrectly interpreted as an edge with no distance, cost, etc.
If formula_7 is an formula_1 matrix containing the edge weights of a graph, then formula_9 (using this distance product) gives the distances between vertices using paths of length at most formula_10 edges, and formula_11 is the distance matrix of the graph.
An arbitrary graph on vertices can be modeled as a weighted complete graph on vertices by assigning a weight of one to each edge of the complete graph that corresponds to an edge of and zero to all other edges. for this complete graph is the adjacency matrix of . The distance matrix of can be computed from as above, however, calculated by the usual matrix multiplication only encodes the number of paths between any two vertices of length exactly . 
Metric distance matrices.
The value of a distance matrix formalism in many applications is in how the distance matrix can manifestly encode the metric axioms and in how it lends itself to the use of linear algebra techniques. That is, if with is a distance matrix for a metric distance, then
When a distance matrix satisfies the first three axioms (making it a semi-metric) it is sometimes referred to as a pre-distance matrix. A pre-distance matrix that can be embedded in a euclidean space is called a Euclidean distance matrix.
Another common example of a metric distance matrix arises in coding theory when in a block code the elements are strings of fixed length over an alphabet and the distance between them is given by the Hamming distance metric. The smallest non-zero entry in the distance matrix measures the error correcting and error detecting capability of the code.
Applications.
Hierarchical clustering.
A distance matrix is necessary for hierarchical clustering.
Phylogenetic analysis.
Distance matrices are used in phylogenetic analysis.
Other uses.
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. They are used in structural and sequential alignment, and for the determination of protein structures from NMR or X-ray crystallography.
Sometimes it is more convenient to express data as a similarity matrix.
It is used to define the distance correlation.
Examples.
For example, suppose these data are to be analyzed, where pixel Euclidean distance is the distance metric.
The distance matrix would be:
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.