Some Investigations On The Ranks And Inertias Of Hermitian Matrix Functions With Applications

It is a very active research topic to investigate matrix equations by theminimal rank or to investigate the definiteness of matrix by the extremal inertiamethod for a long time. It has been extensively studied by a number of scholarsand many useful results have been gotten. In this dissertation, we study someHermitian matrix functions from the above side. As applications, we get manymeaningful new results which extend some known results. In addition, we findthat the rank and the inertia method is one of the important tools for investigatingsome analytical problems such as the stability and the extreme value of somematrix functions. By this results, we investigate such problems for some matrixfunctions correspondingly. These results further enrich and develop the matrixalgebra.The dissertation is divided into 4 chapters.In Chapter 1 the research background as well as the main contributions of thisdissertation are introduced. The basic material for later use are also presented.In Chapter 2 we investigate the extreme ranks and inertias of Hermitianmatrix expressionwhere (X, Y ) is a pair Hermitian solution to AX = B, Y C = D. As applications,we derive the necessary and su?cient conditions for the existence of maximalmatrix ofThe corresponding expressions of the maximal matrix of H is presented whenthe existence conditions are met. In this case, we further prove the matrix func-tion f(X, Y ) is invariable even the pair (X, Y ) changes. Moreover, we estab-lish necessary and su?cient conditions for the system of matrix equationsAX = to have a Hermitian solution and the systemof matrix equations to have a bisymmetric solution. Theexplicit expressions of such solutions to the systems mentioned above are alsoprovided. In addition, we discuss the range of inertias of the matrix functions where X and Y are nonnegative definite pair solutions tosome consistent matrix equations.In Chapter 3, we first consider the maximal and minimal inertias and ranksof the matrix functionswhere, X is a general solution to AX = B, XC = D. Thenwe characterize the real/imaginary part of the classic system of equations AX =B, XC = D. We give the formulas of maximal and minimal ranks and inertiasof and where means the conjugatetranspose and As applications, we further give some necessary andsufficient conditions for the existence of Re-positive (semi)definite, Re-negative(semi)definite, and Re-nonsingular solutions to the system AX = B, XC = D.We also derive the necessary and su?cient conditions for the existence of themaximal matrix and minimal matrix ofG = {P ? MXN±(MXN)?|AX = C,XB = D}.If the conditions are satisfied, the function is stable. In this case, there is only asole element in the matrix set G. The corresponding expression of the element ispresented when the existence conditions are met.In Chapter 4, we construct new quasi quadratic Hermitian structures for asystem of matrix equations. i.e., , where and X is a solution to AX = C, XB = D. We considerthe extremal inertias and ranks of p(X)/q(X). As applications, we derive the necessary and su?cient conditions for p(X)/q(X) is positive (negative), positive(negative) semidefinite, nonsingular and the systemis consistent, respectively. Some special cases such as the unitary solvability andthe contraction solvability for AX = C, XB = D are also considered. In addition,we present the necessary and su?cient conditions for the existence of minimalmatrix of the following Hermitian matrix sets.The explicit expressions of the corresponding minimal matrix is given when theconditions are met. Moreover, we provide algorithms and numerical examples toillustrate our results.