Studies On Some Inverse Problems For Matrices

The inverse problems of matrices (eigenvalues) have extensive applicationsin fields such as automatic control theory, vibration theory, structural design,molecular spectroscopy, parameter identification, network programming,economics, biology, electricity, system engineering, civil structural engineering,solid mechanics, finite elements theory, linear optimal control et al. The inverseproblems for matrices (eigenvalues) mainly include: constrained matrix equationproblems, inverse problems for matrices subject to submatrix constraints, inverseeigenvalue problems for Jacobi matrices, inverse eigenvalue problems for realsymmetric band matrices, and generalized inverse eigenvalue problems formatrices.The constrained matrix equation problems are to find solutions of matrixequations in a set of matrices satisfied some constraint conditions. When theconstraint conditions are different, or the matrix equation is different, we can geta different constrained matrix equation problem.An inverse problem of matrix subject to a submatrix constraint is, undersome constrained conditions, to construct a matrix A with a given matrix A0 asits submatrix.The generalized inverse eigenvalue problem for matrices is an inverseeigenvalue problem of matrix pair subject to spectral restrictions.In my Ph. D. dissertation, the following inverse problems for matrices willbe studied and the corresponding numerical algorithms will also be established.Problem 1 Given X ∈ Rn×m, B ∈ Rn×m, find A ∈ S?Rn×n, such thatf ( A)= AX?B=min.Problem 2 Given Y ∈ Rm×n,X ∈ Rn×p,B ∈ Rm×p, find A ∈ S?Rn×n, suchthatg ( A)= YAX?B=min.Problem 3 Given X ∈ Rn×m,B ∈ Rm×m, find A ∈ S?Rn×n, such thath ( A)= XT AX?B=min.Problem 4 Given A~ ∈Rn×n, find A? ∈SE, such thatAAAAASE? ? ~=m∈i n?~,Here, S is R n× n or a subset of R n× n satisfying some constraint conditionssuch as anti-centrosymmetric, doubly center, doubly center symmetric, generalizedbisymmetric, centrosymmetric, symmetric or anti-symmetric, and ? is Frobeniusnorm, S E denotes the set of solutions for Problem 1 or 2 or 3.Problem 5 Given X ∈ Rn×m, diag(,,,)Λ =λ1 I k1 λ2Ik2LλlIkl,λ 1 , λ2,L ,λlare different, lkmj∑j ==1, find A, B∈S?Rn×n, such thatAX = BXΛ.Problem 6 Given A~ ,B~∈Rn×n, find [ A?, B?]∈SAB, such that[~,~][?,?]inf[~,~][,]ABAB[ ,]ABAB? =A B∈S AB?where { }nnS AB = [ A,B]|AX=BXΛ,A,B∈S?R× denotes the set of solutionsfor Problem 5, S is R n× n or a subset of R n× n satisfying some constraintconditions, such as anti-centrosymmetric, centrosymmetric, symmetricorthogonal symmetric, symmetric orthogonal anti-symmetric, anti-symmetric andskew-symmetric, symmetric and skew anti-symmetric, and ? is Frobeniusnorm.The main results of this dissertation are as follows.1. For Problems 1 and 4 of the constrained matrix equations, we havestudied inverse problems of anti-centrosymmetric, doubly center, doubly centersymmetric and general bisymmetric matrices, and established the methods ofcomputing the corresponding least-squares solutions and the best approximatesolutions. For Problems 2 and 4, we have discussed the optimal approximatesolutions of the inverse problem YAX = Bfor centrosymmetric andanti-centrosymmetric matrices in case of g ( A)=0. For Problems 2 and 3, wehave investigated the least-squares solution of the inverse problem foranti-centrosymmetric matrices. The necessary and sufficient conditions for theexistence of solution have been obtained and its general expression has beenderived. The cooresponding numerical algorithms have been provided.2. Based on the quotient singular value decompositions of the matrix pair,the real matrix extension problem constrained by matrix equation YAX = B, andthe real symmetric and real anti-symmetric matrix extension problemsconstrained by matrix equation X T AX= B have been studied. The optimalapproximation solution of inverse problem under a submatrix constraint, to agiven matrix A~ has also been investigated. The necessary and sufficientcondition for the existence of solution has been established and its expression hasbeen given.3. For Problems 5 and 6, we have discussed the generalized inverseeigenvalue problem for anti-centrosymmetric matrices, and obtained thenecessary and sufficient conditions for existence of a solution. The algorithm andone numerical example for solving optimal approximation solution have beenincluded. For Problem 5, we have investigated the inverse problems forsymmetric orthogonal symmetric, anti-symmetric and skew-symmetric matrices,and obtained the necessary and sufficient conditions for existence of solution.The general form of such solutions has been given and the expression of theoptimal approximation solution to a given matrix has been derived.