A = (a[sub ij]) ∈ R[sup n×n] is termed bisymmetric matrix if a[sub ij] = a[sub ji] = a[sup n ? j + 1, n ? i + 1], i, j = 1, 2 ··· n. We denote the set of all n x n bisymmetric matrices by BSR[sup ...A = (a[sub ij]) ∈ R[sup n×n] is termed bisymmetric matrix if a[sub ij] = a[sub ji] = a[sup n ? j + 1, n ? i + 1], i, j = 1, 2 ··· n. We denote the set of all n x n bisymmetric matrices by BSR[sup n x n]. This paper is mainly concerned with solving the following two problems: Problem I. Given X, B ∈ R[sup n×m], find A ∈ P[sub n] such that AX = B, where P[sub n] = {A ∈ BSR[sup n×n]| x[sup T] Ax ≥ 0, ?x ∈ R[sup n]}. Problem II. Given A[sup *] ∈ R[sup n×n], find ? ∈ S[sub E] such that ||A[sup *] - ?||[sub F] = ... ||A[sup *] - A||[sub F] where || · ||[sub F] is Frobenius norm, and S[sub E] denotes the solution set of problem I. The necessary and sufficient conditions for the solvability of problem I have been studied. The general form of S[sub E] has been given. For problem II the expression of the solution has been provided. [ABSTRACT FROM AUTHOR]展开更多
In this paper, the following problems are considered Problem I. Given A ∈Rm×n, D ∈ R n×n. a) Let S1 = {X: X ∈ Rm×n, ||ATX - XTA - D|| = min} find X ∈ S1 such that ||X|| = min; b) Let S2 = {X: X ∈ R...In this paper, the following problems are considered Problem I. Given A ∈Rm×n, D ∈ R n×n. a) Let S1 = {X: X ∈ Rm×n, ||ATX - XTA - D|| = min} find X ∈ S1 such that ||X|| = min; b) Let S2 = {X: X ∈ Rm×n, ATX - XTA = D} find X ∈ S2 such that ||X|| = min. Problem II. Given A ∈ Rm×n,B B∈Rn×p, D ∈Rm×p. Let L1 = {X: X ∈ SRn×n, AXB = D} find X ∈ L1 such that ||X|| = min. Problem III. Given A ∈ Rm×n,B ∈ Rp×q, C ∈ Rm×q, G ∈ Rl×n, H ∈ Rp×t,D ∈ Rl × t. Let L2 = {X: X ∈ Rn×p, AXB = C, GXH = D} find X ∈ L2 such that ||X|| = min. Using singularvalue and canonical correlation decompositions, the necessary and sufficiellt conditions, under which S2, L1 and L2 are nonempty, are studied. The expressions for the solutions of Problems I, II and III are given.展开更多
文摘A = (a[sub ij]) ∈ R[sup n×n] is termed bisymmetric matrix if a[sub ij] = a[sub ji] = a[sup n ? j + 1, n ? i + 1], i, j = 1, 2 ··· n. We denote the set of all n x n bisymmetric matrices by BSR[sup n x n]. This paper is mainly concerned with solving the following two problems: Problem I. Given X, B ∈ R[sup n×m], find A ∈ P[sub n] such that AX = B, where P[sub n] = {A ∈ BSR[sup n×n]| x[sup T] Ax ≥ 0, ?x ∈ R[sup n]}. Problem II. Given A[sup *] ∈ R[sup n×n], find ? ∈ S[sub E] such that ||A[sup *] - ?||[sub F] = ... ||A[sup *] - A||[sub F] where || · ||[sub F] is Frobenius norm, and S[sub E] denotes the solution set of problem I. The necessary and sufficient conditions for the solvability of problem I have been studied. The general form of S[sub E] has been given. For problem II the expression of the solution has been provided. [ABSTRACT FROM AUTHOR]
文摘In this paper, the following problems are considered Problem I. Given A ∈Rm×n, D ∈ R n×n. a) Let S1 = {X: X ∈ Rm×n, ||ATX - XTA - D|| = min} find X ∈ S1 such that ||X|| = min; b) Let S2 = {X: X ∈ Rm×n, ATX - XTA = D} find X ∈ S2 such that ||X|| = min. Problem II. Given A ∈ Rm×n,B B∈Rn×p, D ∈Rm×p. Let L1 = {X: X ∈ SRn×n, AXB = D} find X ∈ L1 such that ||X|| = min. Problem III. Given A ∈ Rm×n,B ∈ Rp×q, C ∈ Rm×q, G ∈ Rl×n, H ∈ Rp×t,D ∈ Rl × t. Let L2 = {X: X ∈ Rn×p, AXB = C, GXH = D} find X ∈ L2 such that ||X|| = min. Using singularvalue and canonical correlation decompositions, the necessary and sufficiellt conditions, under which S2, L1 and L2 are nonempty, are studied. The expressions for the solutions of Problems I, II and III are given.
