In this paper, we investigate the eigenvalue problem of forward-backward doubly stochastic dii^erential equations with boundary value conditions. We show that this problem can be represented as an eigenvalue problem o...In this paper, we investigate the eigenvalue problem of forward-backward doubly stochastic dii^erential equations with boundary value conditions. We show that this problem can be represented as an eigenvalue problem of a bounded continuous compact operator. Hence using the famous Hilbert-Schmidt spectrum theory, we can characterize the eigenvalues exactly.展开更多
Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute th...Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute the elements of the matrix is derived for a given list ∧ and dependency parameters. Explicit computations are performed for n≤5 and r≤4 to illustrate the result.展开更多
In this paper,we describe how to construct a real anti-symmetric(2p-1)-band matrix with prescribed eigenvalues in its ρ leading principal submatrices.This is done in two steps.First,an anti-symmetric matrix B is cons...In this paper,we describe how to construct a real anti-symmetric(2p-1)-band matrix with prescribed eigenvalues in its ρ leading principal submatrices.This is done in two steps.First,an anti-symmetric matrix B is constructed with the specified spectral data but not necessary a band matrix.Then B is transformed by Householder transformations to a (2ρ-1)-band matrix with the prescribed eigenvalues.An algorithm is presented.Numerical results are presented to demonstrate that the proposed method is effective.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
本文研究如下实对称矩阵广义特征值反问题: 问题IGEP,给定X∈R^(n×m),1=diag(λ_II_k_I,…,λ_pI_k_p)∈R^(n×m),并且λ_I,…,λ_p互异,sum from i=1 to p(k_i=m,求K,M∈SR^(n×n),或K∈SR^(n×n),M∈SR_0^(n×m)...本文研究如下实对称矩阵广义特征值反问题: 问题IGEP,给定X∈R^(n×m),1=diag(λ_II_k_I,…,λ_pI_k_p)∈R^(n×m),并且λ_I,…,λ_p互异,sum from i=1 to p(k_i=m,求K,M∈SR^(n×n),或K∈SR^(n×n),M∈SR_0^(n×m),或K,M∈SR_0^(n×n),或K∈SR^(n×n),M∈SR_+^(n×n),或K∈SR_0^(n×n),M∈SR_+^(n×n),或K,M∈SR_+^(n×m), (Ⅰ)使得 KX=MXA, (Ⅱ)使得 X^TMX=I_m,KX=MXA,其中SR^(n×n)={A∈R^(n×n)|A^T=A},SR_0^(n×n)={A∈SR^(n×n)|X^TAX≥0,X∈R^n},SR_+^(n×n)={A∈SR^(n×n)|X^TAX>0,X∈R^n,X≠0}. 利用矩阵X的奇异值分解和正交三角分解,我们给出了上述问题的解的表达式.展开更多
基金The NSF (10601019 and J0630104) of ChinaChinese Postdoctoral Science Foundation and 985 Program of Jilin University.
文摘In this paper, we investigate the eigenvalue problem of forward-backward doubly stochastic dii^erential equations with boundary value conditions. We show that this problem can be represented as an eigenvalue problem of a bounded continuous compact operator. Hence using the famous Hilbert-Schmidt spectrum theory, we can characterize the eigenvalues exactly.
文摘Given a list of real numbers ∧={λ1,…, λn}, we determine the conditions under which ∧will form the spectrum of a dense n × n singular symmetric matrix. Based on a solvability lemma, an algorithm to compute the elements of the matrix is derived for a given list ∧ and dependency parameters. Explicit computations are performed for n≤5 and r≤4 to illustrate the result.
文摘In this paper,we describe how to construct a real anti-symmetric(2p-1)-band matrix with prescribed eigenvalues in its ρ leading principal submatrices.This is done in two steps.First,an anti-symmetric matrix B is constructed with the specified spectral data but not necessary a band matrix.Then B is transformed by Householder transformations to a (2ρ-1)-band matrix with the prescribed eigenvalues.An algorithm is presented.Numerical results are presented to demonstrate that the proposed method is effective.
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.
