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A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if is a complex symmetric matrix, there is a unitary matrix such that is a real diagonal matrix with non-negative entries.
Synonym for (0,1)-matrix, binary matrix or Boolean matrix. Can be used to represent a k-adic relation. Markov matrix: A matrix of non-negative real numbers, such that the entries in each row sum to 1. Metzler matrix: A matrix whose off-diagonal entries are non-negative. Monomial matrix: A square matrix with exactly one non-zero entry in each ...
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.
For example, that every equivalence relation is symmetric, but not necessarily antisymmetric, is indicated by in the "Symmetric" column and in the "Antisymmetric" column, respectively. All definitions tacitly require the homogeneous relation R {\displaystyle R} be transitive : for all a , b , c , {\displaystyle a,b,c,} if a R b {\displaystyle ...
A symmetric and transitive relation is always quasireflexive. [a] One way to count the symmetric relations on n elements, that in their binary matrix representation the upper right triangle determines the relation fully, and it can be arbitrary given, thus there are as many symmetric relations as n × n binary upper triangle matrices, 2 n(n+1 ...
Given a set and a partial order relation, typically the non-strict partial order , we may uniquely extend our notation to define four partial order relations , <,, and >, where is a non-strict partial order relation on , < is the associated strict partial order relation on (the irreflexive kernel of ), is the dual of , and > is the dual of <.
A square matrix A is called invertible or non-singular if there exists a matrix B such that [28] [29] = =, where I n is the n×n identity matrix with 1s on the main diagonal and 0s elsewhere. If B exists, it is unique and is called the inverse matrix of A , denoted A −1 .
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × ...