MODIFIED NEWTON＇S ALGORITHM FOR COMPUTING THE GROUP INVERSES OF SINGULAR TOEPLITZ MATRICES

Newton＇s iteration is modified for the computation of the group inverses of singular Toeplitz matrices. At each iteration, the iteration matrix is approximated by a matrix with a low displacement rank. Because of the displacement structure of the iteration matrix, the matrix-vector multiplication involved in Newton＇s iteration can be done efficiently. We show that the convergence of the modified Newton iteration is still very fast. Numerical results are presented to demonstrate the fast convergence of the proposed method.     

秩约束子集选择问题的分而治之解法

We consider the rank-constrained subset selection problem(RCSS):Given a matrix A and integer p≤rank(A),fing the largest submatrix A0 consisting of some colmns of A with rank(A0)=p.The RCSS problem is generally NP-hard.This paper focuses on a divide-and -conquer(DC)algorithm for solving the RCSS problem:Partition the matrix A into several small column blocks:A∏=[A1,…,Ak] with a certain column permutation ∏and decompose p to p1 p2 …pk such that solutions of the RCSS problems for smaller couples form a solution of the original RCSS problem.We show that the optimal solution of the RCSS problem can be found by DC algorithm for eachP≤rank (A),if and only if A is column-partitionable,i.e.,rank(A)=∑i=1^k rank(Ai),Based upon QR decomposition,a fast algorithm for determining the column partition is offered. Our divide-and-conquer algorithm is also quite efficient even A is approximately column-partitionable.     

秩减形式的一种新方法（英文）

Let A be an m by n matrix of rank l, and let M and N be m by k and n by q matrices, respectively, where k is not necessarily equal to q or rank(M AN) min(k, q). In this paper, we provide some necessary and sufficient conditions for the validity of the rank subtractivity formula: rank(A-AN(M AN)-M A) = rank(A)-rank(AN(M AN)-M A)by applying the full rank decomposition of A = F G(F ∈ Rm×l, G ∈ Rl×n, rank(A) =rank(F) = rank(G) = l) and the product singular value decomposition of the matrix pair[F M, GN ]. This rank subtractivity formula along with the condition under which it holds is called the extended Wedderburn-Guttman theorem.     

秩$6$的无三角形的图

The rank of a graph G is defined to be the rank of its adjacency matrix A（G）. In this paper we characterize all connected triangle-free graphs with rank 6.     

Relationship Between the Stiffly Weighted Pseudoinverse and Multi-Level Constrained Rseudoinverse

It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, supw∈D‖（W^1/2A) W^1/2‖ = (miniσ (A^(i))^-1, in which (A^(i) is a submatrix of A formed with r = (rank(A)) rows of A, such that (A^(i) has full row rank r. In many practical applications this value is too large to be used. In this paper we consider the case that both A and W(∈D) are fixed with W severely stiff. We show that in this case the weighted pseudoinverse (W^1/2‖A) W^1/2‖ is close to a multilevel constrained weighted pseudoinverse therefore ‖(W^1/2A) W^1/‖2 is uniformly bounded.We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem.     

A PERTURBATION ANALYSIS FOR THE PROJECTION OF A STIFFLY SCALED MATRIX

In this paper we study the perturbation bound of the projection (WA )(WA ), where both the matrices A and W are given with W positive diagonal and severely stiff. When the perturbed matrix A=A δA satisfy several row rank preserving conditions, we derive a new perturbation bound of the projection.     

CSL代数中的有限秩算子 Ⅲ

In terms of the exactly nonzero partition,the reducible projection-system and correlation matrices, two characterizations for a rank three operator in a CSL algebra can be completely decomposed are given.     

On the Range of Certain ASH Algebras of Real Rank Zero*

In this paper, the authors consider the range of a certain class of ASH algebras in [An, Q., Elliott, G. A., Li, Z. and Liu, Z., The classification of certain ASH C*-algebras of real rank zero, J. Topol. Anal., 14(1), 2022, 183–202], which is under the scheme of the Elliott program in the setting of real rank zero C*-algebras. As a reduction theorem, they prove that all these ASH algebras are still the AD algebras studied in [Dadarlat, M. and Loring, T. A., Classifying C*-algebras via ordered, m...     

加边矩阵奇异性的补充

Let consider a m×n matrix A with rank q1 matrix with rank r₁ and H is a r₂×n matrix with rank r₂. Paper [4] gave the necessary and sufficient conditions for the nonsingularity of the bordered matrix M with the case in which r₁=M-q and r₂>n-q . In this short paper, we give the necessary and sufficient conditions for nonsingnlarity of M with the ease in which r₁≥m-q and r₂≥n-q.     

关于幂有限秩算子

An operator F ∈ B(X) is called power finite rank if F~n is of finite rank for some n ∈ N.In this note, we provide several interesting characterizations of power finite rank operators. In particular, we show that the class of power finite rank operators is the intersection of the class of Riesz operators and the class of operators with eventual topological uniform descent.     

图是圈的完全正矩阵

本文证明,图是围的完全正矩阵Ａ当比较矩阵Ｍ（Ａ）的行列式大于零时,恰有两个极小秩１分解,而当ｄｅｔＭ（Ａ）＝０时,恰有一个极小秩１分解．     

关于分块矩阵〔ABCD〕的g-逆的块独立性

本文证明了分块阵M=〔ABCD〕的g-逆有块独立性的充要条件是M适合秩可加性条件,M~+有一特定的表达式的充要条件也是M适合秩可加性条件.本文还给出了包含M的g-逆的子块的不变矩阵以及这些子块的定义方程.     

Stability of certain minimal submanifolds in compact symmetric spaces of rank two

In [T. Kimura, M.S. Tanaka, Totally geodesic submanifold in compact symmetric spaces of rank two, Tokyo J. Math., in press], the authors obtained the global classification of the maximal totally geodesic submanifolds in compact connected irreducible symmetric spaces of rank two. In this paper, we determine their stability as minimal submanifolds in compact symmetric spaces of rank two.     

解一类Hessian矩阵亏秩的修正BFGs算法及其局部Q-超线性收敛性

本文对凸函数在极值点的Hessian矩阵是秩亏一的情况下,给出了一类求解无约束优化问题的修正BFGS算法.算法的思想是对凸函数加上一个修正项,得到一个等价的模型,然后简化此模型得到一个修正的BFGS算法.文中证明了该算法是一个具有超线性收敛的算法,并且把修正的BFGS算法同Tensor方法进行了数值比较,证明了该算法对求解秩亏一的无约束优化问题更有效.     

On a Conjecture of Kontsevich and Variants of Castelnuovo's Lemma

Let A=(aij) be an orthogonal matrix (over R or C) with no entries zero. Let B= (bij) be the matrix defined by bij= 1/ai j. M. Kontsevich conjectured that the rank of B is never equal to three. We interpret this conjecture geometrically and prove it. The geometric statement can be understood as variants of the Castelnuovo lemma and Brianchon's theorem.     

有限域上遍历矩阵的特性研究

对有限域上遍历矩阵的性质进行了分析,给出了有限域上遍历矩阵的计数定理,并对遍历矩阵序对(A,B)关于矩阵M的双侧幂乘集〈A〉M〈B〉的秩及基数进行了全面分析.给出了R_k(A,B)集的构成及其基数的有关定理,所得到的结论对利用遍历矩阵实现有关的公钥密码具有理论上的指导意义.     

Sturm-Liouville systems with rational Weyl functions: Explicit formulas and applications

This paper solves explicitly the direct and inverse spectral problems for Sturm-Liouville systems with rational Weyl functions. As in the previous papers of the authors on canonical and pseudocanonical systems, the method employed in the present paper is based on state space techniques and uses the idea of realization from mathematical system theory. For a subclass of rational potentials a property of modified bispectrality is proven. A wide class of solutions of the matrix Korteweg-de Vries equation is derived.To the memory of M.G. Kreîn, one of the founding fathers of modern spectral theory of differential operators, with admiration and gratitude.     

A type of modified BFGS algorithm with any rank defects and the local Q-superlinear convergence properties

A modified BFGS algorithm for solving the unconstrained optimization, whose Hessian matrix at the minimum point of the convex function is of rank defects, is presented in this paper.The main idea of the algorithm is first to add a modified term to the convex function for obtain an equivalent model, then simply the model to get the modified BFGS algorithm. The superlinear convergence property of the algorithm is proved in this paper. To compared with the Tensor algorithms presented by R. B. Schnabel (seing [4],[5]), this method is more efficient for solving singular unconstrained optimization in computing amount and complication.     

On completions of symmetric and antisymmetric block diagonal partial matrices

A partial matrix is a matrix where only some of the entries are given. We determine the maximum rank of the symmetric completions of a symmetric partial matrix where only the diagonal blocks are given and the minimum rank and the maximum rank of the antisymmetric completions of an antisymmetric partial matrix where only the diagonal blocks are given.