Sign Pattern Matrices and Semirings

Thumbnail Image
Mohindru, Preeti
Journal Title
Journal ISSN
Volume Title
University of Guelph

Sign pattern theory examines what can be said about a matrix if one knows the signs of all or some of its entries but not the exact values. Since all we know is the sign of each entry, we can write these sign patterns as matrices whose entries come from the set {+1, -1, 0, #}, where # is used for an unknown sign. Semirings satisfy all properties of rings with unity except the existence of additive inverses. The set {+1, -1, 0, #} can be viewed as a commutative semiring in natural way. In the thesis, we give a semiring version of the Cayley-Dickson construction which allows one to construct the sign pattern semiring from the Boolean semiring. We use tools from Boolean matrices to study sign nonsingular (SNS) matrices. We also investigate different notions of rank of matrices over semirings. For these rank functions we simplify proofs of classical inequalities for the sum and the product of matrices using the semiring versions of the Cauchy-Binet and Laplace theorems. For matrices over the sign pattern semiring, the minimum rank of the sign pattern is compared with the other versions of the rank. We also characterize irreducible powerful sign pattern matrices and investigate the period and base of an SNS matrix.

Semirings, Sign pattern matrices, Boolean matrices, Ranks