Editing Distance matrix (section)

== Metric distance matrix ==
The value of a distance matrix formalism in many applications is in how the distance matrix can manifestly encode the [[Metric (mathematics)|metric axioms]] and in how it lends itself to the use of linear algebra techniques.  That is, if {{math|1=''M'' = (''x{{sub|ij}}'')}} with {{math|1 ≤ ''i'', ''j'' ≤ ''N''}} is a distance matrix for a metric distance, then

# the entries on the main diagonal are all zero (that is, the matrix is a [[hollow matrix]]), i.e. {{math|1=''x{{sub|ii}}'' = 0}} for all {{math|1 ≤ ''i'' ≤ ''N''}},
# all the off-diagonal entries are positive ({{math|''x{{sub|ij}}'' > 0}} if {{math|''i'' ≠ ''j''}}), (that is, a [[Nonnegative matrix|non-negative matrix]]),
# the matrix is a [[symmetric matrix]] ({{math|1=''x{{sub|ij}}'' = ''x{{sub|ji}}''}}), and
# for any {{mvar|i}} and {{mvar|j}}, {{math|''x{{sub|ij}}'' ≤ ''x{{sub|ik}}'' + ''x{{sub|kj}}''}} for all {{mvar|k}} (the triangle inequality). This can be stated in terms of [[min-plus matrix multiplication|tropical matrix multiplication]]

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]]. For mixed-type data that contain numerical as well as categorical descriptors, [[Gower's distance]] is a common alternative.

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.