一类矩阵方程的双对称定秩解及其最佳逼近

本文研究了矩阵方程AX=B的双对称最大秩和最小秩解问题.利用矩阵秩的方法,获得了矩阵方程AX=B有最大秩和最小秩解的充分必要条件以及解的表达式,同时对于最小秩解的解集合,得到了最佳逼近解.     

矩阵方程AX=B的中心对称定秩解及其最佳逼近

本文研究了矩阵方程AX=B的中心对称解.利用矩阵对的广义奇异值分解和广义逆矩阵,获得了该方程有中心对称解的充要条件以及有解时,最大秩解、最小秩解的一般表达式,并讨论了中心对称最小秩解集合中与给定矩阵的最佳逼近解.     

一类矩阵方程的Hermitian R-对称定秩解(英文)

本文研究了矩阵方程AX=B的Hermitian R-对称最大秩和最小秩解问题.利用矩阵秩的方法,获得了矩阵方程AX=B有最大秩和最小秩解的充分必要条件以及解的表达式,同时对于最小秩解的解集合,得到了最佳逼近解.     

秩约束下矩阵方程AX=B的反对称解及其最佳逼近

本文研究了秩约束下矩阵方程AX=B的反对称解问题.利用矩阵秩的方法,获得了矩阵方程AX=B有最大秩和最小秩解的充分必要条件以及定秩解的表达式,同时对于最小秩解的解集合,得到了最佳逼近解.     

一类矩阵方程的双对称定秩解及其最佳逼近（英文）

本文研究了矩阵方程AX=B的双对称最大秩和最小秩解问题.利用矩阵秩的方法,获得了矩阵方程AX=B有最大秩和最小秩解的充分必要条件以及解的表达式,同时对于最小秩解的解集合,得到了最佳逼近解.     

矩阵Hadamard积和Fan积的特征值界的一些新估计式

本文研究了非奇异M-矩阵A与B的Fan积的最小特征值下界和非负矩阵A与B的Hadamard积 的谱半径上界的估计问题.利用Brauer定理,得到了一些只依赖于矩阵的元素且易于计算的新估计式,改进 了文献[4]现有的一些结果.     

矩阵方程组AX=C,XB=D的公共最小二乘解

通过使用矩阵秩方法,我们给出了矩阵方程组AX =C,XB =D的公共最小二乘解的通解表达式,以及公共最小二乘解的极大秩和极小秩.     

一类四元数矩阵方程的循环解及其最佳逼近

本文研究了四元数体上矩阵方程XB = C 的循环解及其最佳逼近问题. 利用循环矩阵的结构表示式, 以及四元数矩阵的复分解, 得到了方程XB = C 的循环解存在条件及其通解形式; 在循环矩阵约束条件下, 给出了该方程的最小二乘解集合; 与此同时, 在最小二乘解集合中, 获得与给定四元数循环矩阵的最佳逼近解. 推广了约束矩阵方程的数值求解范围. 数值算例验证了本文算法的可行性.     

关于矩阵秩命题的证明

利用齐次线性方程组 AX =0的系数矩阵的秩和它的基础解系之间的关系 ,比较容易地证明许多有关矩阵秩或向量组秩的一些命题 .     

一个矩阵函数及其应用

给出了矩阵函数f（X）=A-BX-（BX）＊的秩和最小惯性指数定理,其中＊表示矩阵的共轭转置.作为应用,给出了Lyapunov矩阵方程以及矩阵不等式BX＋（BX）＊≥A和BX＋（BX）＊≤A可解的若干充要条件.     

A simultaneous decomposition of a matrix triplet with applications

Researches on ranks of matrix expressions have posed a number of challenging questions, one of which is concerned with simultaneous decompositions of several given matrices. In this paper, we construct a simultaneous decomposition to a matrix triplet (A, B, C), where A=±A*. Through the simultaneous matrix decomposition, we derive a canonical form for the matrix expressions A?BXB*?CYC* and then solve two conjectures on the maximal and minimal possible ranks of A?BXB*?CYC* with respect to X=±X* and Y=±Y*. As an application, we derive a sufficient and necessary condition for the matrix equation BXB* + CYC*=A to have a pair of Hermitian solutions, and then give the general Hermitian solutions to the matrix equation. Copyright © 2010 John Wiley & Sons, Ltd.     

Zero cancellation for general rational matrix functions

The problem of cancelling a specified part of the zeros of a completely general rational matrix function by multiplication with an appropriate invertible rational matrix function is investigated from different standpoints. Firstly, the class of all factors that dislocate the zeros and feature minimal McMillan degree are derived. Further, necessary and sufficient existence conditions together with the construction of solutions are given when the factor fulfills additional assumptions like being J-unitary, or J-inner, either with respect to the imaginary axis or to the unit circle. The main technical tool are centered realizations that deliver a sufficiently general conceptual support to cope with rational matrix functions which may be polynomial, proper or improper, rank deficient, with arbitrary poles and zeros including at infinity. A particular attention is paid to the numerically-sound construction of solutions by employing at each stage unitary transformations, reliable numerical algorithms for eigenvalue assignment and efficient Lyapunov equation solvers.     

How to use RSVD to solve the matrix equation A=BXC

Through the restricted singular value decomposition (RSVD) of the matrix triplet (C, A, B), we show in this note how to choose a variable matrix X such that the matrix pencil A ? BXC attains its maximal and minimal ranks. As applications, we show how to use the RSVD to solve the matrix equation A = BXC.     

On Ranks of Matrices Associated with Trees

A sign pattern matrix is a matrix whose entries are from the set {+,–,0}. The purpose of this paper is to obtain bounds on the minimum rank of any symmetric sign pattern matrix A whose graph is a tree T (possibly with loops). In the special case when A is nonnegative with positive diagonal and the graph of A is star-like, the exact value of the minimum rank of A is obtained. As a result, it is shown that the gap between the symmetric minimal and maximal ranks can be arbitrarily large for a symmetric tree sign pattern A. Supported by NSF grant No. DMS-00700AMS classification: 05C50, 05C05, 15A48     

Least‐squares solutions and least‐rank solutions of the matrix equation AXA* = B and their relations

A Hermitian matrix X is called a least‐squares solution of the inconsistent matrix equation AXA^* = B, where B is Hermitian. A^* denotes the conjugate transpose of A if it minimizes the F‐norm of B ? AXA^*; it is called a least‐rank solution of AXA^* = B if it minimizes the rank of B ? AXA^*. In this paper, we study these two types of solutions by using generalized inverses of matrices and some matrix decompositions. In particular, we derive necessary and sufficient conditions for the two types of solutions to coincide. Copyright © 2012 John Wiley & Sons, Ltd.     

Possible numbers of ones in the squares of 0–1 matrices with a given rank

For a symmetric 0–1 matrix A, we give the number of ones in A ² when rank(A) = 1, 2, and give the maximal number of ones in A ² when rank(A) = k (3 ≤ k ≤ n). The sufficient and necessary condition under which the maximal number is achieved is also obtained. For generic 0–1 matrices, we only study the cases of rank 1 and rank 2.     

Least squares solutions with special structure to the linear matrix equation AXB = C

In this article we give some formulas for the maximal and minimal ranks of the submatrices in a least squares solution X to AXB = C. From these formulas, we derive necessary and sufficient conditions for the submatrices to be zero and other special forms, respectively. Finally, some Hermitian properties for least squares solution to matrix equation AXB = C are derived.