Some Results On Matrix Sparsity, Matrix Partial Ordering And Binary Rank

We study some problems on symmetric7-matrices, binary ranks and binary factor-izations of nonnegative integer matrices, matrix partial orderings and reverse order law of generalized inverses, partial orderings and reverse order law of generalized inverses of elements of C*-algebras, an iterative method for computing the Drazin inverse and the minimum rank of regular class of binary matrices. Our main results are as follows.1. We study the structural and sparsity properties of symmetric7-matrices. We deter-mine the largest number of zero entries of a symmetric7-matrix and determine the symmetric7-matrices that attain this largest number.2. We investigate the binary ranks and binary factorizations of nonnegative integer matrices. We give sharp bounds on the binary rank, determine the matrices that attain the lower bound, establish some necessary conditions for a nonnegative integer matrix to achieve the upper bound. We also compute the binary ranks of nonnegative integer matrices with low ranks and small orders, estimate the binary ranks of several special classes of nonnegative integer matrices.3. Some results relating different matrix partial orderings and the reverse order law for generalized inverses are given. Special attention is paid when one of the two involved matrices is EP.4. We establish some results relating star, left-star, right-star, minus ordering and the reverse order law for generalized inverses of elements of C*-algebras.5. We construct a new iterative method for computing the Drazin inverse and deduce a necessary and sufficient condition for its convergence to the Drazin inverse. We also present an error bound for the iterative method for approximating the Drazin inverse. 6. We study the minimum rank of regular class of binary matrices, prove that v(n, k)=[n/k]+k if and only if n (?)±1(mod k) and thus answer a question posed by Pullman and Stanford. Furthermore, a necessary condition for v(n, k)=[n/k]+k is given.