On Matrix Completions And Nonnegative Matrices

There are five parts in this thesis. In the first part, we discuss the completion of a partial integral matrix to a unimodular matrix; in the second part, we investigate the possible numbers of positive entries of imprimitive nonnegative matrices; in the third, we consider upper bounds for the spectral radius of the Hadamard product of nonnegative matrices and lower bounds for the minimum eigenvalue of the Fan product of M-matrices; In the fourth, we consider the general solution formula for D in Davis, Kahan, and Weinberger'snorm-preserving extension theorem; finally we consider the unique completion of a partial positive semidefinite matrix.1. Completions of a partial integral matrix to a unimodular matrixWe prove that if a partial integral matrix has a free diagonal then this matrix can be completed to a unimodular matrix. Such a condition is necessary in a general sense. Consequently if an n×n (n≥2) partial integral matrix has 2n- 3 prescribed entries and any n entries of these do not constitute a row or a column then it can be completed to a unimodular matrix. This improves a recent result of Zhan.2. Possible numbers of positive entries of imprimitive nonnegative matricesIn [31], Zhan has determined the maximum and minimum numbers of positive entriesof imprimitive irreducible nonnegative matrices with a given imprimitivity index. Letσ(A, k) denote the number of positive entries of an n x n irreducible nonnegative matrix A with imprimitivity index k. Let M(n, k) and m(n, k) denote the maximum and minimum numbers of positive entries of imprimitive irreducible nonnegative matrices of order n with a given imprimitivity index k, respectively. In a seminar, Zhan once asked whether or not when m,(n, k)≤d≤M(n, k) where d is a positive integer, then we can find annxn irreducible nonnegative matrix A with imprimitivity index k such that d =σ(A, k). We answer this question affirmatively. 3. Bounds on eigenvalues of the Hadamard product and the Fan product of matricesWe prove an upper bound for the spectral radius of the Hadamard product of nonnegativematrices and a lower bound for the minimum eigenvalue of the Fan product of M-matrices. These improve two existing results.4. The general solution formula for W in Kahan's theoremIn 1967, Kahan obtained a matrix extension theorem: Suppose H∈C^l×l is Hermi-tian and B∈C^s×l. Denote the spectral norm of R = (?)by‖R‖₂. Then thereexists a W∈C^s×s such that A = (?) is Hermitian and‖A‖₂=‖R‖₂. Kahandid not give an explicit expression for W. In [11], Davis, Kahan, and Weinberger had generalized Kahan's matrix extension theorem. They proved that there exist solutions D toand indicated how to construct all solutions D.Let (?) =‖R‖₂. In [34, 35], Zheng showed that one may takeFurthermore, the inequalitygives the general solution formula for W in Kahan's theorem. The generalized inverse form solution of Kahan's matrix theorem is new. In this thesis, we will give the generalizedinverse form solution of Davis, Kahan, and Weinberger's norm-preserving extension theorem. Consequently, we can get Zheng's result and our proof is simpler than that of Zheng.5. The unique completion of a partial positive semidefinite matrix In this thesis the problem of unique completion of a partial positive semidefinite matrixto a positive semidefinite matrix is considered. We give a sufficient and necessary condition for partial tridiagonal positive semidefinite matrices to have unique positive semidefinite completion. We also consider the unique positive semidefinite completion of partial positive semidefinite matrices whose associated graphs are chordal graphs. Finallywe pose a conjecture which characterizes those partial positive semidefinite matrices which have unique positive semidefinite completion.