Binary matrix

In mathematics, particularly matrix theory, a binary matrix or (0,1)-matrix is a matrix in which each entry is either zero or one. For example:

: is a 2 × 2 binary matrix.

Frequently operations on binary matrices are defined in terms of modular arithmetic mod 2 — that is, the elements are treated as elements of the Galois field GF(2) = $mathbb\{Z\}\_2$. They arise in a variety of representations and have a number of more restricted special forms.

The number of "m×n" binary matrices is equal to 2^mn, and is thus finite.

Examples

Examples of binary matrices are numerous:

* A permutation matrix is a (0,1)-matrix, all of whose columns and rows each have exactly one nonzero element.
** A Costas array is a special case of a permutation matrix
* An incidence matrix in combinatorics and finite geometry has ones to indicate incidence between points (or vertices) and lines of a geometry, blocks of a block design, or edges of a graph (mathematics)
* A design matrix in analysis of variance is a (0,1)-matrix with constant row sums.
* An adjacency matrix in graph theory is a matrix whose rows and columns represent the vertices and whose entries represent the edges of the graph. The adjacency matrix of a simple, undirected graph is a binary symmetric matrix with zero diagonal.
* The biadjacency matrix of a simple, undirected bipartite graph is a (0,1)-matrix, and any (0,1)-matrix arises in this way.
* The prime factors of a list of "m" square-free, "n"-smooth numbers can be described as a "m"×π("n") (0,1)-matrix, where π is the prime-counting function and "a"_"ij" is 1 if and only if the "j"th prime divides the "i"th number. This representation is useful in the quadratic sieve factoring algorithm.