Indicator vector explained

In mathematics, the indicator vector, characteristic vector, or incidence vector of a subset T of a set S is the vector

x _{T} : = (x _{s)} _{s\in} such that x _{s} = 1 if s \in T and x _{s} = 0 if s \notin T.

If S is countable and its elements are numbered so that

S = \{s _{1,s} _{2,\ldots,s} _{n\}} , then x _{T} = (x _{1,x} _{2,\ldots,x} _{n)} where x _{i} = 1 if s _{i} \in T and x _{i} = 0 if s _{i} \notin T.

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.^{[1] [2] [3]}

An indicator vector is a special (countable) case of an indicator function.

Example

If S is the set of natural numbers

N , and T is some subset of the natural numbers, then the indicator vector is naturally a single point in the 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.

Notes and References

1. Book: Mirkin, Boris Grigorʹevich . Mathematical Classification and Clustering. 112. 0-7923-4159-7. 1996. 10 February 2014.
2. A Tutorial on Spectral Clustering. Ulrike. von Luxburg. Ulrike von Luxburg . Statistics and Computing. 17. 4. 2007. 2. 10 February 2014. 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].pdf. 6 February 2011. dead.
3. Book: Taghavi, Mohammad H. . Decoding Linear Codes Via Optimization and Graph-based Techniques. 21. 2008. 9780549809043 . 10 February 2014.