A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n+1-j,n+1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular val...A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n+1-j,n+1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A^TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.展开更多
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
文摘A real n×n symmetric matrix X=(x_(ij))_(n×n)is called a bisymmetric matrix if x_(ij)=x_(n+1-j,n+1-i).Based on the projection theorem,the canonical correlation de- composition and the generalized singular value decomposition,a method useful for finding the least-squares solutions of the matrix equation A^TXA=B over bisymmetric matrices is proposed.The expression of the least-squares solutions is given.Moreover, in the corresponding solution set,the optimal approximate solution to a given matrix is also derived.A numerical algorithm for finding the optimal approximate solution is also described.
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]
