# Logical matrix

A **logical matrix**, **binary matrix**, **relation matrix**, **Boolean matrix**, or **(0,1) matrix** is a matrix with entries from the Boolean domain **B** = {0, 1}. Such a matrix can be used to represent a binary relation between a pair of finite sets.

## Contents

## Matrix representation of a relation[edit]

If *R* is a binary relation between the finite indexed sets *X* and *Y* (so *R* ⊆ *X*×*Y*), then *R* can be represented by the logical matrix *M* whose row and column indices index the elements of *X* and *Y*, respectively, such that the entries of *M* are defined by:

In order to designate the row and column numbers of the matrix, the sets *X* and *Y* are indexed with positive integers: *i* ranges from 1 to the cardinality (size) of *X* and *j* ranges from 1 to the cardinality of *Y*. See the entry on indexed sets for more detail.

### Example[edit]

The binary relation *R* on the set {1, 2, 3, 4} is defined so that *aRb* holds if and only if *a* divides *b* evenly, with no remainder. For example, 2*R*4 holds because 2 divides 4 without leaving a remainder, but 3*R*4 does not hold because when 3 divides 4 there is a remainder of 1. The following set is the set of pairs for which the relation *R* holds.

- {(1, 1), (1, 2), (1, 3), (1, 4), (2, 2), (2, 4), (3, 3), (4, 4)}.

The corresponding representation as a Boolean matrix is:

## Other examples[edit]

- 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 (discrete 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. - A bitmap image containing pixels in only two colors can be represented as a (0,1)-matrix in which the 0's represent pixels of one color and the 1's represent pixels of the other color.
- A binary matrix can be used to check the game rules in the game of Go
^{[1]}

## Some properties[edit]

The matrix representation of the equality relation on a finite set is an identity matrix, that is, one whose entries on the diagonal are all 1, while the others are all 0.

If the Boolean domain is viewed as a semiring, where addition corresponds to logical OR and multiplication to logical AND, the matrix representation of the composition of two relations is equal to the matrix product of the matrix representations of these relation. This product can be computed in expected time O(*n*^{2}).^{[2]}

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) = ℤ_{2}. They arise in a variety of representations and have a number of more restricted special forms. They are applied e.g. in XOR-satisfiability.

The number of distinct *m*-by-*n* binary matrices is equal to 2^{mn}, and is thus finite.

## See also[edit]

Wikimedia Commons has media related to .Binary matrix |

- List of matrices
- Binatorix (a binary De Bruijn torus)
- Redheffer matrix
- Relation algebra

## Notes[edit]

**^**Petersen, Kjeld (February 8, 2013). "Binmatrix". Retrieved August 11, 2017.**^**Patrick E. O'Neil, Elizabeth J. O'Neil (1973). "A Fast Expected Time Algorithm for Boolean Matrix Multiplication and Transitive Closure" (PDF).*Information and Control*.**22**(2): 132–138. doi:10.1016/s0019-9958(73)90228-3. — The algorithm relies on addition being idempotent, cf. p.134 (bottom).

## References[edit]

- 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, Binary Matrices - Kim, Ki Hang,
*Boolean Matrix Theory and Applications*, ISBN 0-8247-1788-0

## External links[edit]

- Hazewinkel, Michiel, ed. (2001) [1994], "Logical matrix",
*Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4