Some Matrix Equations Problems Under Rank-constrained And Their Optimal Approximation Problems

The constrained matrix equations have a wide practical background in controltheory,vibration theory,computational physics and nonlinear programming.This Ph.D. thesis considers the following problems systematically:Problem I Given matrices A?Cm×n and B?Cm×n, and a positive integers. Find X?S ? Cn×n, such that AX = B, and r(X) = s. In the case S1 = {X?S|AX = B} is nonempty, find (?)X?S1 r(X), M? = Xm?aSx1 r(X),and determine the minimal rank solutions in S1, that is (?)Problem II Given matrices A?Rp×m, B?Rn×q, C?Rp×n, and D?Rm×q,and a positive integer s. Find X?Rm×n such that AX = C,XB = D, andr(X) = s. In the case where the solution set S1 = {X?Rm×n|AX = C,XB = D}is nonempty, find (?)X?S1 r(X),M = Xm?aSx1 r(X),and determine the minimal rank solutions in S1, i. e., Sm? = {X | r(X) = m?,X?S1}.Problem III Given matrices X?Rn×m and B?Rm×m. Find A?S ? Rn×nsuch that XTAX = B,Problem IV Given X??Rn×n. Find X??(?) such that (?)Here S(?) is the solution set of problem I, II or III and ||·|| is the Frobenius normof matrices, S is Rn×n or a subset of Cn×n satisfying some constraint conditions.The major contributions of this thesis includes1. When S is chosen to be CRn×n(P), Can×n(P) , CSRn×n , ACSRn×n, re-spectively, by the use of the generalized singular value decomposition (GSVD) ofmatrices, we derive solution set of the matrix equation AX = B with prescribedrank. We also obtain the optimal approximation solution to the set of the minimalrank solution S(?).2. When S is chosen to be SRn×n ,ASRn×n, respectively, by the use of matrixrank theory, we give a deep discussion to the rank of the symmetric and anti-symmetric solution of the matrix equation AX = B. We then obtain the optimal approximation solution to the minimal rank. When S is chosen to be BSRn×n orBASRn×n, respectively, by the use of the expression theorem of these two classesof matrices, we adopt the decomposition technique to derive the necessary andsuffcient condition for the solvability of Problem I. We also get the expression of thegeneral solution for Problem I. In addition, we obtain the optimal approximationsolution to the set of the minimal rank solution S().3. Early in [61], the solvability and the expression of the general equationof matrix equations have been well studied. With the help of minimum semi-norm g-inverse, the authors in [61] studied the prescribed rank solution of thematrix equation AX = C, XB = D. We note that [61] did not consider thesolution of Problem VI. Indeed, the technique used in [61] seems not enough toderive the solution of Problem VI. In This thesis, unlike the idea in [61], we usethe matrix decomposition technique and rank theory to solve Problems II andVI successfully. We ?rst give the expression of the solution with prescribed rankthrough generalized Moore-Penrose inverse. We then drive the expression of thesolution with minimal rank via matrix block decomposition technique and matrixrank theory. The result obtained in the thesis is the only best approximationsolution to Problem VI.4. We adopt the generalized singular-value decomposition technique to studythe solutions of matrix equation XTAX = B subject to anti-centro-symmetricand anti-symmetric ortho-symmetric matrix. This problem comes from vibratingtheory. We also study the related best approximation problem. We derive someconditions for the existence of the problem. We also get the expressions for thegeneral solution of the problem.This Ph.D. Thesis are supported by the Natural Science Foundation of Chinaand the Doctorate Foundation of the Ministry of Education of China.This Ph.D. Thesis is typeset by software LATEX2?.