Hermite-广义反Hamilton矩阵的一类反问题

给定矩阵X和B,利用矩阵的广义奇异值分解,得到了矩阵方程X~HAX=B有Hermite-广义反Hamiton解的充分必要条件及有解时解的—般表达式.用S_E表示此矩阵方程的解集合,证明了S_E中存在唯一的矩阵(?),使得(?)与给定矩阵A的差的Frobenius范数最小,并且给出了矩阵(?)的表达式;同时也证明了S_E中存在唯一的矩阵A_o,使得A_o是此矩阵方程的极小Frobenius范数Hermite-广义反Hamilton解,并且给出了矩阵A_o的表达式.

One Kind of Inverse Problem for the Hermite-generalized Antihamilton Matrices

In this paper,we first consider the solution of the matrix equation as follows: find a hermite-generalized antihamilton matrix such that for given matrices X,B we have X~HAX=B.By using generalized singular value decomposition of matrices,the necessary and sufficient conditions for the existence of and the expressions for the solutions of the matrix equation are derived.We denote the set of such solutions by S_E.Then the matrix nearness problem and the solution of minimum Frobenius norm for the matrix equation are discussed.That are:Given an arbitrary A~*,find a matrix A(?)S_E which is nearest to A~* in Frobenius norm,and find a matrix A_0(?)S_E which is the matrix of minimum Frobenius norm.We show that the nearest matrix and the matrix of minimum Frobenius norm are unique and provide expressions for this matrices.