Indicator vector

In mathematics, the indicator vector, characteristic vector, or incidence vector of a subset T of a set S is the vector ${\displaystyle x_{T}:=(x_{s})_{s\in S}}$ such that ${\displaystyle x_{s}=1}$ if ${\displaystyle s\in T}$ and ${\displaystyle x_{s}=0}$ if ${\displaystyle s\notin T.}$

If S is countable and its elements are numbered so that ${\displaystyle S=\{s_{1},s_{2},\ldots ,s_{n}\}}$, then ${\displaystyle x_{T}=(x_{1},x_{2},\ldots ,x_{n})}$ where ${\displaystyle x_{i}=1}$ if ${\displaystyle s_{i}\in T}$ and ${\displaystyle x_{i}=0}$ if ${\displaystyle s_{i}\notin T.}$

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 ${\displaystyle \mathbb {N} }$, 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

1. Mirkin, Boris Grigorʹevich (1996). Mathematical Classification and Clustering. p. 112. ISBN 0-7923-4159-7. Retrieved 10 February 2014.
2. von Luxburg, Ulrike (2007). "A Tutorial on Spectral Clustering" (PDF). Statistics and Computing. 17 (4): 2. Archived from the original (PDF) on 6 February 2011. Retrieved 10 February 2014.
3. Taghavi, Mohammad H. (2008). Decoding Linear Codes Via Optimization and Graph-based Techniques. p. 21. ISBN 9780549809043. Retrieved 10 February 2014.