Results On Combinatorial And Analytic Properties Of Matrices

We study some problems on 0-1 matrices, partial matrices, sign patterns and nonnegative matrices. Our main results are as follows.1. By using techniques from graph theory and matrix theory, we prove that the number of entry 1's in a 0-1 matrix A is less than or equal to n(n-1)/2 if Ak is also a 0-1 matrix, where k≥n-1 is a given positive integer, and that the maximum number is attained if and only if A is permutation similar to the strictly upper triangular matrix all of whose entries above the diagonal are ones.2. We determine the maximum number of entries equal to 1 in those 0-1 matrices of order n all of whose powers are 0-1 matrices as well as the matrices that attain this maximum number, where the maximum number is (n+1)2/4 if n is odd and n(n+2)/4 if n is even.3. We characterize the partial matrices all of whose completions have the same determinant.4. We characterize the partial matrices all of whose completions have the same rank and determine the maximum number of indeterminates in such partial matrices of a given order as well as the matrices that attain this maximum number. This work is closely related to the open problem of determining the minimum rank of all completions of a given partial matrix.5. We characterize the nilpotent sign patterns of a given index k and deter-mine the maximum number of nonzero entries in such sign patterns as well as the sign patterns that attain this maximum number. This work is related to the Turan graph. 6. We prove the inequalities where the matrices are nonnegative, AoB denotes the Hadamard product of A and B,‖A‖_∞andρ(A) denote the spectral norm and spectral radius of A, respectively. The first inequality improves a classical inequality of Schur, and the second one generalizes a result of Zhan-Audenaert.7. We solve two conjectures on singular values and unitarily invariant norms of nonnegative matrices negatively.Part of this dissertation is joint work with Professor Richard A. Brualdi and Professor Xingzhi Zhan.