Research On Several Classes Of Constrained Matrix Equations And Their Fixed Rank Solutions

The constrained matrix equation problem has been widely used in many fields such as system identification, structural design, automatic control theory, vibration theory, linear optimal control, and finite elements and so on. Solving constrained matrix equations for fixed rank solutions has great significance to perfect the theory of constrained matrix equation.The following problems are considered systematically in this M.S. thesis:(?)Determine M, m and give the representations of the elements in S 0 and the optimal approximation to a given element. (?)Determine M, m and give the representations of the elements in S m and the optimal approximation to a given element. (?)Determine the maximal or minimal rank of element (?) Determine m and give the representations of the elements in S mf . The main achievements are as follows: (1) For Problem I , the maximal and minimal rank of element X in S1 and the representations of the elements in S 0 are obtained by using singular value decomposition, matrix partition and relevant theories of rank. The optimal approximate solution to a given element is obtained.(2) For problemⅡ, by mainly using singular value decomposition, the quotient singular value decomposition and relevant theories of rank , we obtain the maximal and minimal ranks of element X in S E, the representations of the elements in S m are obtained.(3) For problem III and problem IV, these are solved by mainly using generalized singular value decomposition and relevant theories of rank .