The Least Square Solutions With The Minimum-norm For Some Matrix Equations

The problem of solving the linear matrix equation is one of the important researchfields of the numerical linear algebra. It has been widely applied in biology, electricity,molecular spectroscopy, vibration theory, finite elements, structural design, solid me-chanics, parameter identification, automatic control theory, linear optimal control andso on.This thesis is mainly concerned with the problem about how to get the least-squares solutions of some matrix equations. These problems are as follows:Problem I Given A∈R^m×n,B∈R^n×p,C∈R^m×p, LetS_E = {X|X∈S,||AXB - C|| = min},find (X|^)∈S_E, such thatwhere S is the set of centro-symmetric matrices or centro-skew symmetric matrices.Problem II Given A∈R^m×p,B∈R^n×p,C∈R^p×q,D∈R^p×l,E∈R^m×q,F∈R^n×l,G∈R^m×l, LetS_E = {X|X∈R^p×p, ||(AXC,BXD,AXD) - (E,F,G)|| = min},find (X|^)∈S_E, such thatProblem III Given A∈R^m×p,B∈R^n×p,C∈R^p×q,D∈R^p×l,E∈R^m×q,F∈R^n×l,G∈R^m×l,H∈R^n×q, LetS_E = {X|X∈R^p×p, ||(AXC,BXD,AXD,BXC) - (E,F,G,H)|| = min},find (X|^)∈S_E, such thatProblem IV Given A∈R^m×p,B∈R^n×p,E∈R^m×m,F∈R^n×n,G∈R^m×n,Let S_E = {X|X∈S, ||(AXA^T,BXB^T,AXB^T) - (E,F,G)|| = min},find (X|^)∈S_E, such thatwhere S is the set of symmetric matrices or skew-symmetric matrices.Problem V Given A∈R^m×p,B∈R^n×p,E∈R^m×m,F∈R^n×n,G∈R^m×n,H∈R^n×m, LetS_E = {X|X∈S, ||(AXA^T,BXB^T,AXB^T,BXA^T) - (E,F,G,H)|| = min},find (X|^)∈S_E, such thatwhere S is the set of symmetric matrices or skew-symmetric matrices.In problems I to V, ||·|| is Frobenius norm, S (?) R^n×n satisfies some constraintconditions, such as symmetric matrix, skew-symmetric matrix, centro-symmetric ma-trix, centro-skew symmetric matrix and so on.In this thesis , using the projection theorem, we could get the following results:1. When S is the set of centro-symmetric matrices or centro-skew symmetric ma-trices, we get the least square-solutions with the minimum-norm of the matrix equationAXB = C;2. We get the least square-solutions with the minimum-norm of the matrix equa-tions (AXC,BXD,AXD) = (E,F,G), (AXC,BXD,AXD,BXC) = (E,F,G,H);3. When S is the set of symmetric matrices or skew-symmetric matrices, we get theleast square-solutions with the minimum-norm of the matrix equations (AXA^T,BXB^T,AXB^T) = (E,F,G), (AXA^T,BXB^T,AXB^T,BXA^T) = (E,F,G,H) .The least square solution got by the GSVD or CCD can't fill with the otho-invariable of Frobenius , so we can't find out the minimum norm least square solutionsdirectly. The advantage of projection theorem in this thesis is that by connectingthe GSVD with CCD, it has successfully overcame the traditional di?culties above.The thesis transforms the minim norm problem to the answer of consistence equa-tion(equations) by the projection theorem, so that a series of problems mentionedabove can be solved.