Boolean matrix

In mathematics, a Boolean matrix is a matrix with entries from a Boolean algebra. When the two-element Boolean algebra is used, the Boolean matrix is called a logical matrix. (In some contexts, particularly computer science, the term "Boolean matrix" implies this restriction.)

Let U be a non-trivial Boolean algebra (i.e. with at least two elements). Intersection, union, complementation, and containment of elements is expressed in U. Let V be the collection of n × n matrices that have entries taken from U. Complementation of such a matrix is obtained by complementing each element. The intersection or union of two such matrices is obtained by applying the operation to entries of each pair of elements to obtain the corresponding matrix intersection or union. A matrix is contained in another if each entry of the first is contained in the corresponding entry of the second.

The product of two Boolean matrices is expressed as follows: $(AB)_{ij}=\bigcup _{k=1}^{n}(A_{ik}\cap B_{kj}).$

According to one author, "Matrices over an arbitrary Boolean algebra β satisfy most of the properties over β₀ = {0, 1}. The reason is that any Boolean algebra is a sub-Boolean algebra of $\beta _{0}^{S}$ for some set S, and we have an isomorphism from n × n matrices over $\beta _{0}^{S}\ {\text{to}}\ \beta _{n}^{S}.$ "^[1]

References

^ Ki-Hang Kim (1982) Boolean Matrix Theory and Applications, page 249, Appendix: Matrices over arbitrary Boolean Algebras, Marcel Dekker ISBN 0-8247-1788-0

R. Duncan Luce (1952) "A Note on Boolean Matrices", Proceedings of the American Mathematical Society 3: 382–8, Jstor link MR0050559
Jacques Riguet (1954) "Sur l'extension du calcul des relations binaires au calcul des matrices à éléments dans une algèbre de Boole", Comptes Rendus 238: 2382–2385

Stan Gudder & Frédéric Latrémolière (2009) "Boolean inner-product spaces and Boolean matrices", Linear Algebra and Its Applications 431: 274–96 MR2522576
D.E. Rutherford (1963) "Inverses of Boolean matrices", Proceedings of the Glasgow Mathematical Association 6: 49–63 MR0148585
T.S. Blythe (1967) "Eigenvectors of Boolean Matrices", Proceedings of the Royal Society of Edinburgh 67: 196–204 MR0210727
Steven Kirkland & Norman J. Pullman (1993) "Linear Operators Preserving Invariants of Non-binary Boolean Matrices", Linear and Multilinear Algebra 33: 295–300 doi:10.1080/03081089308818200 MR1334678
Kyung-Kae Kang, Seok-Zun Song & Young-Bae Jung (2011) "Linear Preservers of Regular Matrices over General Boolean Algebras", Bulletin of the Malaysian Mathematical Sciences Society, second series, 34(1): 113–25 MR2783783

v t e Matrix classes
Explicitly constrained entries	Alternant Anti-diagonal Anti-Hermitian Anti-symmetric Arrowhead Band Bidiagonal Bisymmetric Block-diagonal Block Block tridiagonal Boolean Cauchy Centrosymmetric Conference Complex Hadamard Copositive Diagonally dominant Diagonal Discrete Fourier Transform Elementary Equivalent Frobenius Generalized permutation Hadamard Hankel Hermitian Hessenberg Hollow Integer Logical Matrix unit Metzler Moore Nonnegative Pentadiagonal Permutation Persymmetric Polynomial Quaternionic Signature Skew-Hermitian Skew-symmetric Skyline Sparse Sylvester Symmetric Toeplitz Triangular Tridiagonal Vandermonde Walsh Z
Constant	Exchange Hilbert Identity Lehmer Of ones Pascal Pauli Redheffer Shift Zero
Conditions on eigenvalues or eigenvectors	Companion Convergent Defective Definite Diagonalizable Hurwitz-stable Positive-definite Stieltjes
Satisfying conditions on products or inverses	Congruent Idempotent or Projection Invertible Involutory Nilpotent Normal Orthogonal Unimodular Unipotent Unitary Totally unimodular Weighing
With specific applications	Adjugate Alternating sign Augmented Bézout Carleman Cartan Circulant Cofactor Commutation Confusion Coxeter Distance Duplication and elimination Euclidean distance Fundamental (linear differential equation) Generator Gram Hessian Householder Jacobian Moment Payoff Pick Random Rotation Routh-Hurwitz Seifert Shear Similarity Symplectic Totally positive Transformation
Used in statistics	Centering Correlation Covariance Design Doubly stochastic Fisher information Hat Precision Stochastic Transition
Used in graph theory	Adjacency Biadjacency Degree Edmonds Incidence Laplacian Seidel adjacency Tutte
Used in science and engineering	Cabibbo–Kobayashi–Maskawa Density Fundamental (computer vision) Fuzzy associative Gamma Gell-Mann Hamiltonian Irregular Overlap S State transition Substitution Z (chemistry)
Related terms	Jordan normal form Linear independence Matrix exponential Matrix representation of conic sections Perfect matrix Pseudoinverse Row echelon form Wronskian
Mathematics portal List of matrices Category:Matrices

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