In this paper, we consider the following two problems: Problem i. Given X ∈ Rmxn,A = diag（λ1,…, λm） > 0, find A E BSR such that where ||AX-X∧||=min, is Frobenius norm, BSR: is the set of all n x n bisymmetri...In this paper, we consider the following two problems: Problem i. Given X ∈ Rmxn,A = diag（λ1,…, λm） > 0, find A E BSR such that where ||AX-X∧||=min, is Frobenius norm, BSR: is the set of all n x n bisymmetric nonnegative definite matrices. Problem Ⅱ. Given A* ∈ Rnxn, find ALS ∈ SE such that||A*-ALS||=inf||A*-A|| where SE is the solution set of problem Ｉ. The existence of the solution for problem Ⅰ, Ⅱ and the uniqueness of the solution for Problem Ⅱ are proved. The general form of SE is given and the expression of ALS is presented.展开更多
Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk...Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.展开更多
In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a g...In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.展开更多
Least squares solution of F=PG with respect to positive semidefinite symmetric P is considered,a new necessary and sufficient condition for solvablity is given,and the expression of solution is derived in the some spe...Least squares solution of F=PG with respect to positive semidefinite symmetric P is considered,a new necessary and sufficient condition for solvablity is given,and the expression of solution is derived in the some special cases. Based on the expression, the least spuares solution of an inverse eigenvalue problem for positive semidefinite symmetric matrices is also given.展开更多
文摘In this paper, we consider the following two problems: Problem i. Given X ∈ Rmxn,A = diag（λ1,…, λm） > 0, find A E BSR such that where ||AX-X∧||=min, is Frobenius norm, BSR: is the set of all n x n bisymmetric nonnegative definite matrices. Problem Ⅱ. Given A* ∈ Rnxn, find ALS ∈ SE such that||A*-ALS||=inf||A*-A|| where SE is the solution set of problem Ｉ. The existence of the solution for problem Ⅰ, Ⅱ and the uniqueness of the solution for Problem Ⅱ are proved. The general form of SE is given and the expression of ALS is presented.
基金Project 19771020 supported by National Science Foundation of China.
文摘Let T1,n be an n x n unreduced symmetric tridiagonal matrix with eigenvaluesand is an (n - 1) x (n - 1) submatrix by deleting the kth row and kth column, k = 1, 2,be the eigenvalues of T1,k andbe the eigenvalues of Tk+1,nA new inverse eigenvalues problem has put forward as follows: How do we construct anunreduced symmetric tridiagonal matrix T1,n, if we only know the spectral data: theeigenvalues of T1,n, the eigenvalues of Ti,k-1 and the eigenvalues of Tk+1,n?Namely if we only know the data: A1, A2, An,how do we find the matrix T1,n? A necessary and sufficient condition and an algorithm ofsolving such problem, are given in this paper.
基金This work was supported by The National Natural Science Foundation of China, under grant 10271074.
文摘In this paper, we discuss an inverse eigenvalue problem for constructing a 2n × 2n Jacobi matrix T such that its 2n eigenvalues are given distinct real values and its leading principal submatrix of order n is a given Jacobi matrix. A new sufficient and necessary condition for the solvability of the above problem is given in this paper. Furthermore, we present a new algorithm and give some numerical results.
文摘Least squares solution of F=PG with respect to positive semidefinite symmetric P is considered,a new necessary and sufficient condition for solvablity is given,and the expression of solution is derived in the some special cases. Based on the expression, the least spuares solution of an inverse eigenvalue problem for positive semidefinite symmetric matrices is also given.
