Some Special Classes Of Tridiagonal Matrix And The Matrix Equation X ~ Of Tax = B Matrix Inverse Problem

Inverse problems of matrices have been widely used in control theory, economic field, vibration theory and civil struction engineering. We will systematically study some inverse problems of matrices in this M.S thesis We mainly discuss the following problems:Problem I Given x,y,z,~v ER" Sc R~'~,Find A ~ S such thatA(x,y) =(:,w)Problem II GivenX ~, B ~ R""m Sc R"~", FindA G Ssuch thatk.~T4X — B~ = mm or '('TAY = B.The main results of this paper are as follows:When S is respectively(1)unseparately real symmetric and skew-symmetric(we call it bisvmmetric) tridiagonal matrices(:)unseparately real symmetric and skew-antisvmmetric tridiagonal matrices. (3) unseparately real antisvmmetric tridiagonai matrices;(4)unseparately real antisymmetric and skew-symmetric tridiagonal matrices.(5)unseparately real antisymmetric and skew-antisymmetric (we call it bi-antisvmmetric ) tridiagonal matrices;We have firstly discussed the construction of this kind of matrices.And then we have proved the sufficient (or sufficient and necessary) conditions when problem I have solutions or unique solution. The expression of the solutions and numeral examples are given separately.2 'When S is respectively(1)centrally symmetric matrices;(2)bisymmetric matrices;(3)bisymmetric and semidefinate positive matrices;We have obtained some necessary and sufficient conditions for solvability of problem II.When the conditions are satisfied, the expression of the solutions are given.