For an ordered list 
in a metric space with metric 
, the distance matrix of 
is the square matrix
 |
(1)
|
It is symmetric, has zero diagonal and nonnegative entries, and its entries obey the triangle inequality 
. Conversely, any matrix with these
properties and with 
iff 
defines a metric on its index set. A dissimilarity matrix
need not satisfy all of these conditions.
Suppose 
,
..., 
are points in a Euclidean space. The matrix of their squared distances,
 |
(2)
|
is commonly called a squared Euclidean distance matrix, or sometimes simply a Euclidean distance matrix. Let 
be the all-ones vector and define
 |  |  |
(3)
|
 |  |  |
(4)
|
Then 
is the gram matrix of the points after translating
their centroid to the origin. A symmetric matrix 
with zero diagonal is a squared
Euclidean distance matrix iff 
is positive semidefinite;
equivalently, 
whenever 
(Schoenberg 1935). The smallest Euclidean dimension in which the distances can be
realized is 
(Young and Householder 1938). If 
contains unsquared Euclidean distances, these criteria
are applied to 
.
A graph distance matrix is the distance matrix obtained from the graph distances among the vertices of a graph. Distance matrices also serve as input to multidimensional scaling and other methods that reconstruct or approximate point configurations from pairwise distances.
The distance matrix of a list of objects can be computed in the Wolfram Language using DistanceMatrix[
x1, ..., xn
], with the metric specified by the DistanceFunction
option when needed.
