Hadamard Products Of Matrices And Sign Patterns

In this thesis, we mainly consider some problems for Hadamard products and Fan products of nonnegative matrices and M-matrices. Some inequalities of Hadamard productsof matrices are given. At the same time, some special matrices such as inverse M-matrices, zero-pattern invariant matrices, sign pattern matrices, fc-potent matrices and sign k-potent matrices are studied. These results are related to some work of M. Fiedler, R.A. Horn, R. Mathias, R. Bhatia, C. Davis, M.D. Choi, C. Eschenbach, M. Jeter and W. Pye.1. Hadamard products of nonnegative matricesLet A = (a_ij) and B = (b_ij) be nonnegative matrices, D₁ = diag(a_ii) and D₂ = diag(b_ii). We give a sharp upper bound on the spectral radiusρ(A (?) B). In particular, if all diagonal entries of A and B are nonzero, thenwhere J_A = D₁^-1l(A - D₁) and J_B = D₂^-1(B - D₂).2. Hadamard products and Fan products of M-matricesLet A = (a_ij) and B = (b_ij) be nonsingular M-matrices, D₁ = diag(a_ii) and D₂ = diag(b_ii). A sharp lower bound on the smallest eigenvalueÏ„(A *B) for the Fan product of A and B is given, and a sharp lower bound onÏ„(A(?)B^-1) for the Hadamard product of A and B^-1 is derived as follows:andwhere S_A = D₁^-1(D₁- A) and S_B = D₂^-1(D₂ - B).3. Some norm inequalities for Hadamard products of matrices Let C_n and R_n⁺ denote the sets of n×n complex and nonnegative real matrices respectively,and‖·‖_F be the Frobenins norm. We first characterize those unitarily invariant norms‖·‖satisfying‖[a_ij]‖=‖[|a_ij|]‖for all [a_ij]∈C_n. We then prove that if‖·‖is a norm on C_n, thenfor all X∈R_n⁺,A,B∈C_n if and only if the norm ||·|| satisfiesand that if A, B, X∈C_n, thenfor any unitarily invariant norm‖·‖, where |A| = (A^* A)^1/2. These results are related to some work of R.A. Horn and R. Mathias and work of R. Bhatia, C. Davis and M.D. Choi.4. Inverse M-matrices and zero-pattern invariant matricesIt is known that an inverse M-matrix is a nonnegative zero-pattern invariant matrix. In this thesis, we first characterize the structures of inverse M-matrices by considering zero-pattern invariant matrices. We next consider nonnegative matrices with at most three nonzero entries in each row, and obtain a sufficient and necessary condition for such a matrix to be an inverse M-matrix. Thus we also obtain a sufficient and necessary conditionfor a triadic matrix A to be an inverse matrix. In particular, if such a matrix A is also a (0, 1)-matrix, then A is an inverse M-matrix if and only if A is nonsingular and zero-pattern invariant. In addition, some properties of Hadamard products of inverse M-matrices are given.5. Sign idempotent pattern matricesIt is shown that not all sign idempotent patterns are similar to nonnegative sign patterns.We present two classes of sign idempotent patterns that are similar to nonnegative sign patterns. An open problem posed by C. Eschenbach is answered. 6. K-potent matrices with negative entriesLet K(A) be the set of all real matrices with the same sign pattern as a real matrix A. We characterize the structure of a k-potent real matrix A with the property that X^k+1∈K(A) for every matrix X∈K(A), where A has no zero rows or columns. Further, we identify sign k-potent matrices which allow k-potence. Thus, an open problem posed by C. Eschenbach is answered as our corollary.