Some Results On Special Matrices And Nonlinear Matrix Equations

In this dissertation, we study some problems on the nonlinear matrix equations Xs+A*X-tA=Q and Xs+A*F(X)A=Q; the norm inequalities for an accretive-dissipative matrix; the spectral radius of the Hadamard product of two nonnegative matrices; the minimum eigenvalue of the Fan product of two M-matrices; the minimum eigenvalue of the Hadamard product of M-matrices and the inverse of M-matrices; the properties of diagonally magic matrices; perturbation bounds for eigenvalues of diagonalizable matrices and perturbation bounds for singular values of arbitrary matrices. Our main results are as follows.1. Investigations of the nonlinear matrix equation Xs+A*X-tA=Q, where Q is a Hermitian positive definite matrix. We consider four cases of this equation:(a) s and t are positive integers, we present the upper bound of [det(AA*)]1/n when the nonlinear matrix equation Xs+A*X-tA=Q has a Hermitian positive def-inite solution. We also get the bounds of [detX]1/n and trX for the existence of a Hermitian positive definite solution. We obtain some bounds for the eigenval-ues of the Hermitian positive definite solution. We derive tight bounds of the partial sum and partial product about the eigenvalues of the solution X for the nonlinear matrix equation Xs+A*X-tA=Q;(b) s≥1,0<t≤1, necessary conditions for the existence of a Hermitian positive definite solution is given;(c)0<s≤1,t≥1, necessary conditions for the existence of a Hermitian positive definite solution is also given;(d) s,t>0, we present some properties of the Hermitian positive definite solutions. We also get a property of the spectral radius of A for the existence of a solution. The spectral norm of A for the existence of a Hermitian positive definite solution is given.2. Investigations of the nonlinear matrix equation Xs+A*F(X)A=Q with s≥1. Several sufficient and necessary conditions for the existence and uniqueness of the Hermitian positive (semidefinite) definite solution are derived; the fixed point itera-tion is given; and perturbation bounds are presented.3. We present inequalities between the norm of off-diagonal blocks and the norm of diagonal blocks of an accretive-dissipative matrix, and inequalities between the norm of the whole accretive-dissipative matrix and the norm of its diagonal blocks.4. A new upper bound on the spectral radius for the Hadamard product of A and B is obtained, where A and B are nonnegative matrices. Meanwhile, a new lower bound on the smallest eigenvalue for the Fan product of A and B is got, and some new lower bounds on the minimum eigenvalue for the Hadamard product of B and A-1are given, where A and B are nonsingular M-matrices.5. A new class of matrices called diagonally magic matrices is presented and studied. In particular, we prove that every diagonally magic matrix has rank at most2; every square submatrix of a diagonally magic matrix is still a diagonally magic matrix; the products of diagonally magic matrices with some given matrices are diagonally magic matrices.6. Perturbation bounds for the eigenvalues of diagonalizable matrices are derived. Per-turbation bounds for singular values of arbitrary matrices are also given.