Minimum matrix rank of k-regular (0,1) matrices

On generalized Hadamard matrices of minimum rank

Generalized Hadamard matrices of order qn−1 (q a prime power, n ≥ 2) over GF (q) are related to symmetric nets in affine 2-(qn, qn−1, (qn−1 − 1)/(q − 1)) designs invariant under an elementary abelian group of order q acting semi-regularly on points and blocks. The rank of any such matrix over GF (q) is greater than or equal to n− 1. It is proved that a matrix of minimum q-rank is unique up to a...

Diagonal entry restrictions in minimum rank matrices

Let F be a field, let G be a simple graph on n vertices, and let S (G) be the class of all F -valued symmetric n × n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. For each graph G, there is an associated minimum rank class, M R (G) consisting of all matrices A ∈ S (G) with rankA = mr (G), the minimum rank among all matrices in S (G)....

E-Clean Matrices and Unit-Regular Matrices

Let $a, b, k,in K$ and $u, v in U(K)$. We show for any idempotent $ein K$, $(a 0|b 0)$ is e-clean iff $(a 0|u(vb + ka) 0)$ is e-clean and if $(a 0|b 0)$ is 0-clean, $(ua 0|u(vb + ka) 0)$ is too.

Commuting maps on rank-k matrices

GENERALIZED REGULAR FUZZY MATRICES

In this paper, the concept of k-regular fuzzy matrix as a general- ization of regular matrix is introduced and some basic properties of a k-regular fuzzy matrix are derived. This leads to the characterization of a matrix for which the regularity index and the index are identical. Further the relation between regular, k-regular and regularity of powers of fuzzy matrices are dis- cussed.