The Solutions Of Several Classes Of Linear Matrix Equations And The Procrustes Problems

The unconstrained and constrained linear matrix equations and related least squares(L-S) problems have been of interest for many applications, including particle physics and geology, control theory, the inverse Sturm-Liouville problem, inverse problems of vibration theory, digital image and signal processing, photogrammetry, finite elements and multidimensional approximation.Suppose A, B, C, D be given, A be a diagonal matrix, X, Y be unknown matrices, and letBy using of a series of methods in numerical linear algebra, such as the singular value decomposition (SVD) and the polar decomposition(PD) of a matrix, the generalized singular value decomposition (GSVD), the quotient singular value decomposition (QSVD) and the canonical correlation decomposition (CCD) of a pair of matrices, the generalized inverse of a matrix, the vector operator and the Kronecker product, the dual space theory and approximation principle in Hilbert space, the linear manifold and so on, this Ph.D. thesis has solved a lot of problems as follows.1. In this thesis, the necessary and sufficient conditions for the existence of and the gerenal expressions for the symmetrizable, semidefinite-symmetrizable andpositive-definite-symmetrizable solutions of the equation f1 (X) = 0 are obtained,and also, the related least-squares problems are solved.2. In this thesis, the necessary and sufficient conditions for the existence of and the gerenal expressions the symmetric, skew-symmetric and semi-definitesolutions of the equation f2 (X) = 0 are obtained, and also, the general solutions of the least-squares problems f2 (X) = mm over the symmetric matrices set(SR mm),the skew-symmetric matrices set (ARmm) and the semi-definite matrices set R0mm are derived. When the solution is unique, the logarithm is given.3. The minimal norm solutions of the equation f2(X) = 0 over Rmm, SRmm and ARmm are derived by using the CCD method for the first time.4. The minimal norm solutions of the equation f4(X) = 0 over SRmxm are derived, and also the necessary and sufficient conditions for the existence of and the general expressions of the skew-symmetric solution of the equation f4 (X) = 0are obtained, the related L-S problem f4 (X) = min is solved.5. The necessary and sufficient conditions for the existence of and the general expressions of the skew-symmetric solution of the generalized Sylvester equationf5 (X, Y) = 0 are obtained, and the related optimal approximation solution isobtained, too. And also the necessary and sufficient conditions for the existence of and the general expressions of the permutation symmetric solution of the equationf6(X,Y) = 0 are obtained, the related optimal approximation problem is solved.6. The necessary and sufficient conditions for the existence of and the general expressions of the skew-symmetric solution of the Lyapunov equation f7 (X, Y) = 0are obtained, and the related optimal approximation problem is solved.At last, on several linear manifolds, the conditions for the existence of and the general expressions of the symmetric, skew-symmetric and symmetric semi-definitesolution of the linear matrix equation f3 (X) = 0 are discussed. And the relatedoptimal approximation solutions are obtained.This Ph.D. thesis is supported by the National Natural Science Foundation of China.