Some Matrix Equations Problems Under Special Constraint And Their Optimal Approximation Problems

The constraint matrix equation problems are the problems which are to findsolutions to a matrix equation in a constraint matrix set. The study of constraintmatrix equation not only has great significance for the study of matrix theory andmethods, but also has many applications in science and technology computing,such as control theory, information theory, vibration theory, system identification,structural dynamic model updating and automated system simulation. This dis-sertation considers the following constraint matrix equation problems.Problem?Given A, B?Rp×n, C, D?Rn×q, X0?Rk×k, find X?S, such that minX?S( AX - B F2 + XC -D F2 )where S is the set of symmetric, anti- symmetric, skew-symmetric or skew anti-symmetric matrices, and any X?S satisfies X[1 : k] = X0.Problem?GivenA?Rm×(2p+k), B?R(2p+k)×q, C?Rm×q, X0?Rk×k,findX?S,such that min X?S AXB -C Fwhere S is the set of centrosymmetric, centroskew symmetric, bisymmetric,anti-bisymmetri, symmetric and skew anti-symmetric or anti-symmetric and skew sym-metric matrices, and any X?S satisfies Xc(k) = X0.Problem?Let SE is the solution set to problem and problem givenmatrix X?Rn×n,findX?SE,such thatX -X F = Xm?iSnE X - X FProblem?aGivenA, B?Cm×nfindX?Crn×n(P, Q)(Can×n(P, Q)),such thatAX = BProblem?GivenA, B, C?Cn×n,findX?Crn×n(P, Q)(Can×n(P, Q)),such thatAXB = CThe main results of this dissertation are as follows:1. The matrix form LSQR algorithms, to find the leading submatrix con-straint symmetric, anti-symmetric, skew-symmetric, skew anti-symmetric solutions and its approximate solutions of the matrix equations (AX = B, XC = D), areestablished.2. The matrix form LSQR algorithms, to find the center submatrix constraintcenter symmetric(center antisymmetric), bisymmetric(antibisymmetric), symmet-ric and skew antisymmetric, antisymmetric and skew symmetric solutions and itsapproximate solutions of the matrix equations AXB = C, are established.3. The necessary and su?cient conditions for the solvability of and the ex-pression for the generalized (P, Q) re?exive(anti-re?exive) solutions to the matrixequations AX = B and AXB = C are obtained. Furthermore, the expression ofthe approximate generalized (P, Q) re?exive(anti-re?exive) solution to the matrixequation AX = B is also made clear in this dissertation.The main innovation points of this dissertation are summarized as follows.1. It is first time to find leading submatrix solutions of matrix equations byLSQR algorithm combined with block matrix techniques.2. Least squares problem of matrix equation under principal submatrix con-straint is turned into least squares problem of matrix equations in terms of thestructure nature of constraint matrix so as to reduce the error accumulation ap-pearing in the process of iteration by the way of lessening calculation of the samestructure block, which has never been studied before.3.Using vec operator,least squares problem of high-order vector is turned intoleast squares problem of low-order Matrix,so lessen the burden of memory in theprocess of iteration.4.Initial values are skillfully chosen, so as to obtain minimal norm least squaresconstraint solution to matrix equation with principal submatrix .5. The abundant numerical examples in the dissertation give authenticationto the e?ectiveness of matrix form LSQR algorithm.6.The generalized (P, Q)re?exive (anti-re?exive) re?exive solutions of matrixequations are studied in terms of block matrix techniques.This dissertation is typeset by software LATEX2?.