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Concept
Logical matrix
Summary
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. It is an important tool in combinatorial mathematics and theoretical computer science.
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 article on indexed sets for more detail.
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, 2R4 holds because 2 divides 4 without leaving a remainder, but 3R4 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 logical matrix is
which includes a diagonal of ones, since each number divides itself.
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.
A design matrix in analysis of variance is a (0, 1)-matrix with constant row sums.
A logical matrix may represent an adjacency matrix in graph theory: non-symmetric matrices correspond to directed graphs, symmetric matrices to ordinary graphs, and a 1 on the diagonal corresponds to a loop at the corresponding vertex.
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