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:

${\begin{pmatrix}0&1\\1&0\\\end{pmatrix}}$ 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 jth prime divides the ith number. This representation is useful in the quadratic sieve factoring algorithm.

See also

• List of matrices
• Redheffer matrix

References

• Hogben, Leslie (2006), Handbook of Linear Algebra (Discrete Mathematics and Its Applications), Boca Raton: Chapman & Hall/CRC, ISBN 978-1-58488-510-8, section 31.3