The Special Solution To Some Matrix Equations And Their Optimal Approximation

The row symmetric solution and generalized reflexive solution to matrix equations are widely used in system identification, technology computing and so on. Based on these two solutions of matrix equations, we studied the generalized row symmetric matrix has the properties of generalized reflexive matrix and row symmetric matrix. This dissertation considers the following problems.Problem Ⅰ Given the mother matrix X1∈RPQn×n of X, consider the generalized row symmetric of X when it has even rows.Problem Ⅱ Given the mother matrix X1∈RPQn×n of X, consider the generalized row symmetric of X when it has odd rows.Problem Ⅲ The generalized row symmetric solution of matrix equation AX=B and its optimal approximation.Problem Ⅳ The generalized row symmetric solution of matrix equation AX+BY=C and its optimal approximation, where X,Y∈Crn×n(P,Q)×Can×n(P,Q).Problem Ⅴ The optimal generalized row symmetric approximation solution of matrix equation AX=B when a block of X is constrained.The main results of this dissertation are as follows:1.Using the properties of generalized row symmetric matrix, we can prove that X is a generalized row symmetric matrix if it has even rows, then find out the conditions should be satisfied if X is a generalized row symmetric matrix with odd rows.2.The sufficient and necessary conditions of the generalized row symmetric solutions of matrix equation AX=B when X has even rows and odd rows and the expression of solution, then got the optimal approximation when the matrix equation is not tolerate. Find out the optimal generalized row symmetric approximation solution when the leading submatrix is constraint.3.The sufficient and necessary condition of the generalized row symmetric solution to matrix equation AX+BY=C and the expression of solution be obtained, when the matrix equation is not tolerate, the optimal approximation solution be also obtained, where X,Y∈Crn×n(P,Q)×Can×n(P,Q).