Definition:Logical Matrix

Definition

A logical matrix is a matrix whose entries are all either $0$ or $1$.

Also known as

A logical matrix is also known as a boolean matrix, for George Boole.

Examples

Loop-Digraph

The adjacency matrix for a loop-digraph is a logical matrix:

$\qquad \begin{pmatrix}

1 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ 1 & 0 & 1 & 0 \\ \end{pmatrix}$

Relation

A logical matrix can be used as an explicit definition of a binary relation.

The binary relation is a divisor of on the set of natural numbers $\set {1, 2, 3, 4}$ consists of the set of ordered pairs:

$\set {\tuple {1, 1}, \tuple {1, 2}, \tuple {1, 3}, \tuple {1, 4}, \tuple {2, 2}, \tuple {2, 4}, \tuple {3, 3}, \tuple {4, 4} }$

This relation can be represented by the following logical matrix:

$\begin {pmatrix}

1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end {pmatrix}$

Also see

• Definition:Irreducible Logical Matrix

• Results about logical matrices can be found here.