# Indicator vector

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In mathematics, the '''indicator vector''', '''characteristic vector''', or '''incidence vector''' of a [subset](/source/subset) ''T'' of a [set](/source/Set_(mathematics)) ''S'' is the vector <math>x_T := (x_s)_{s\in S}</math> such that <math>x_s = 1</math> if <math>s \in T</math> and <math>x_s = 0</math> if <math>s \notin T.</math>

If ''S'' is [countable](/source/countable_set) and its elements are numbered so that <math>S = \{s_1,s_2,\ldots,s_n\}</math>, then <math>x_T = (x_1,x_2,\ldots,x_n)</math> where <math>x_i = 1</math> if <math>s_i \in T</math> and <math>x_i = 0</math> if <math>s_i \notin T.</math>

To put it more simply, the indicator vector of ''T'' is a vector with one element for each element in ''S'', with that element being one if the corresponding element of ''S'' is in ''T'', and zero if it is not.<ref>{{cite book|title=Mathematical Classification and Clustering|first= Boris Grigorʹevich |last=Mirkin|page=112|isbn=0-7923-4159-7|year=1996|publisher= Springer |url=https://books.google.com/books?id=brzLe4X4ypEC&dq=indicator+vector+subset&pg=PA170|access-date=10 February 2014}}</ref><ref>{{cite journal|title=A Tutorial on Spectral Clustering|first=Ulrike|last=von Luxburg|author-link= Ulrike von Luxburg |journal=Statistics and Computing|volume=17|issue=4|year=2007|page=2|url=http://www.kyb.mpg.de/fileadmin/user_upload/files/publications/attachments/Luxburg07_tutorial_4488%5B0%5D.pdf|access-date=10 February 2014|archive-url=https://web.archive.org/web/20110206100855/http://www.kyb.mpg.de/fileadmin/user_upload/files/publications/attachments/Luxburg07_tutorial_4488%5B0%5D.pdf#91;0&#93;.pdf|archive-date=6 February 2011|url-status=dead}}</ref><ref>{{cite book|title=Decoding Linear Codes Via Optimization and Graph-based Techniques|first=Mohammad H. |last=Taghavi|page=21|year=2008|isbn=9780549809043 |url=https://books.google.com/books?id=6UCpDYih3WcC&dq=%22indicator+vector%22+subset&pg=PA21|access-date=10 February 2014}}</ref>

An indicator vector is a special (countable) case of an [indicator function](/source/indicator_function).

==Example==
If ''S'' is the set of [natural number](/source/natural_number)s <math>\mathbb{N}</math>, and ''T'' is some subset of the natural numbers, then the indicator vector is naturally a single point in the [Cantor space](/source/Cantor_space): that is, an infinite sequence of 1's and 0's, indicating membership, or lack thereof, in ''T''.  Such vectors commonly occur in the study of [arithmetical hierarchy](/source/arithmetical_hierarchy).

==Notes==
{{reflist}}

Category:Basic concepts in set theory
Category:Vectors (mathematics and physics)

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