This paper studies the following two problems: Problem I. Given X, B is-an-element-of R(n x m), find A is-an-element-of P(s,n), such that AX = B, where Ps, n = {A is-an-element-of SR(n x n)\x(T) Ax greater-than-or-equ...This paper studies the following two problems: Problem I. Given X, B is-an-element-of R(n x m), find A is-an-element-of P(s,n), such that AX = B, where Ps, n = {A is-an-element-of SR(n x n)\x(T) Ax greater-than-or-equal-to 0, for-all S(T) x = 0, for given S is-an-element-of R(p)n x p}. Problem II. Given A* is-an-element-of R(n x n), find A is-an-element-of S(E), such that \\A*-A\\ = inf(A is-an-element-of S(E) \\A*-A\\ where S(E) denotes the solution set of Problem I. The necessary and sufficient conditions for the solvability of Problem I, the expression of the general solution of Problem I and the solution of Problem II are given for two cases. For the general case, the equivalent form of conditions for the solvability of Problem I is given.展开更多
Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ...Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn（F）, Mn（F）}, we say that a linear map f from G1 to G2 is inverse-preserving if f（X）^-1 = f（X^-1） for every invertible X ∈ G1. Let L （G1, G2） denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L（Sn（F）,Mn（F））, L（Sn（F）,Sn（F））, L （Mn（F）,Mn（F）） and L（Mn （F）, Sn （F）） are characterized.展开更多
文摘This paper studies the following two problems: Problem I. Given X, B is-an-element-of R(n x m), find A is-an-element-of P(s,n), such that AX = B, where Ps, n = {A is-an-element-of SR(n x n)\x(T) Ax greater-than-or-equal-to 0, for-all S(T) x = 0, for given S is-an-element-of R(p)n x p}. Problem II. Given A* is-an-element-of R(n x n), find A is-an-element-of S(E), such that \\A*-A\\ = inf(A is-an-element-of S(E) \\A*-A\\ where S(E) denotes the solution set of Problem I. The necessary and sufficient conditions for the solvability of Problem I, the expression of the general solution of Problem I and the solution of Problem II are given for two cases. For the general case, the equivalent form of conditions for the solvability of Problem I is given.
基金supported in part by the Chinese Natural Science Foundation under Grant No.10271021the Natural Science Foundation of Heilongjiang Province under Grant No.A01-07the Fund of Heilongjiang Education Committee for Overseas Scholars under Grant No.1054
文摘Suppose F is a field different from F2, the field with two elements. Let Mn（F） and Sn（F） be the space of n × n full matrices and the space of n ×n symmetric matrices over F, respectively. For any G1, G2 ∈ {Sn（F）, Mn（F）}, we say that a linear map f from G1 to G2 is inverse-preserving if f（X）^-1 = f（X^-1） for every invertible X ∈ G1. Let L （G1, G2） denote the set of all inverse-preserving linear maps from G1 to G2. In this paper the sets .L（Sn（F）,Mn（F））, L（Sn（F）,Sn（F））, L （Mn（F）,Mn（F）） and L（Mn （F）, Sn （F）） are characterized.
