On The Rank Range Of The Least-squares Solutions Of Several Classes Of Matrix Equations

The least-squares solutions of constrained matrix equation have been widelyused in many felds such as structural design,system identifcation and linear controland so on.Researching on the least-squares solutions of constrained matrix equa-tion have great consequence to the theory and practical application of constrainedmatrix equation.This M.S thesis considers the following problems:Problem1:GivenA∈C^m×p,B∈C^m×n,suppose X∈S Cp×n,such that∥AX B∥=min.（1）Determine the largest rank M and the minimum rank m of the least-squaressolutions;（2）Give respectively expression of the least-squares solutions with the largestrank and the minimum rank;（3）If the integer t satisfy m <t <M,give the least-squares solutions of rankt.Problem2:Given A∈C^m×n, B∈C^p×q, C∈C^m×q, suppose S is the set of theleast-squares solutions of AXB=C, promptlyS={X∈Cn×p|∥AXB C∥=min},（1）Seek the rank range [m, M] of S;（2）If t∈[m, M],suppose the set St S denote the set with rank t of elementsin S, seek the general expression of elements in St.Problem3:GivenA∈R^m×n,B∈R^m×m,suppose X∈SR^n×n,such that AXAT B=min.（1）Determine the largest rank M and the minimum rank m of the symmetryleast-squares solutions;（2）Give respectively expression of the symmetry least-squares solutions withthe largest rank and the minimum rank;（3）If the integer t satisfy m <t <M,give the symmetry least-squares solutionsof rank t.The main achievements of this thesis includes:1, In problem1, when S is chosen to be Cp×n, by using the singular valuedecomposition of matrices, QR decomposition and relevant theories of rank for partitioned matrix,we derive the largest rank,the minimum rank and the so-lution set of arbitrary rank.2, When S is chosen to be SR^n×n,ASR^n×n,BSR^n×n,or KSR^n×n,respectively,weobtain the rank of the least-squares solutions of the matrix equation AX=B,bythe use of the property of block matrix,singular value decomposition and relevanttheories of rank.We also derive the least-squares solutions with fxed rank.3, For problem2,by the use of singular value decomposition, elementary trans-formation and relevant property of rank,we obtain the range of rank and the rep-resentation of the elements in St.4, For problem3,we adopt the property of symmetric matrix and singular valuedecomposition of matrix to study the rank of least-squares solutions.We also getthe particular solutions with fxed rank.