Some Researches For Special Matrices

With the rapid development of the application displines for the special matrices in numerical analysis, optimization theory, automatic control, system identification and en-gineering calculations, it is becoming a focus on the matrix theory and the numerical algebra to further study the special matrices and matrix equations on special matrix sets which have been derived more and more attentions.In this thesis, some special equations and matrix equations on some special matrix sets, inverse eigenvalue problems of two special matrices and the matrix inverse comple-tion problems are investigated. It mainly includes:(1) By using the relationship between (P, Q) orthogonal symmetric matrices which are newly defined and symmetric matrices, projection theorem in inner space, the gen-eral expressions for the least square (P,Q-orthogonal symmetric solutions, least square optimal approximate solution, least square with minimum-norm solution of the matrix equation ATXB=C are obtained. Some numerical results have been given.(2) Inverse eigenproblems for Hermitian-Hamilton matrices and symmetric matri-ces with submatrix constraint of KX=MX A in structural dynamic model updating are studied with the help of their special structures and the base method, and the solvability conditions and the general expressions of solutions for these two inverse problems are derived.(3) By using the Moore-Penrose generalized inverse and its related projectors, based on [55,87], the solvability of the matrix equations AX+XTB=C and AXB+CXTD=E are discussed, and some solvability conditions are derived, then the solvability for the operator equation AX+X*B=C is studied, at last the solution of a complex least squares with constrained phase of AX=B is given.(4) By using the relationship between roots and coefficients of quadratic polynomial and its properties, the necessary and sufficient conditions for the equienergetic equation of two special matrices are given.(5) The necessary and sufficient conditions for the following two kinds of inverse completion problems are obtained. Thus, some of the problems proposed in [93] are solved. (a) Given A=±A*∈Cm×m, B=±B*∈Cp×p,C∈Cm×p,find X∈Cm×p such that(b) Given A∈Cn×m,B∈Cn×q, C∈Cp×m,G1∈Cm×p,G2∈Cq×n,G3∈Cn×m, where n+p=m+q,find X∈Cp×p such thatFurthermore, to our knowledge, it is still an open problem to find an explicit formula for the Drazin of if A-1 exists and the Schur complement D-CA-1B is singular. By using the reverse order law of the Drazin inverse, the conditions for the existence and one kind of expressions of MD in the open problem are given, and some representations for the Drazin inverse of M under rank equality constraints are also ob-tained.(6) By using the Brualdi theorem, two inequalities for the spectral radius of Hadamard product of the nonnegative matrices are obtained, thus the problems proposed in [75] are solved.