实循环矩阵与实对角矩阵相似问题

本文中我们证明了与实对角矩阵相似的每一个实循环矩阵都是对称的．并给出了一个正交变换，使得任意的ｎ×ｎ实循环对称矩阵通过该变换与实对角矩阵相似．     

半正定分块矩阵和一个线性矩阵方程及其反问题

一个实的（未必对称）ｎ×ｎ矩阵Ａ称为半正定的，如果对任意非零的ｎ维行向量ｘ，均有ｘＭｘ^t≥０．本文给出了一个分块ｎ×ｎ矩阵为半正定的充要条件．另外，我们讨论了线性矩阵方程ＡＸ＝Ｂ对解附加种种条件下的解．我们应用矩阵在相抵下的标准形给出了这一方程的相容性的充要条件．还给出这个方程的反问题在对解附加各种条件下的解．     

反对称正交对称矩阵反问题

本文讨论一类反对称正交对称矩阵反问题及其最佳逼近．研究了这类矩阵的一些性质，利用这些性质给出了反问题解存在的一些条件和解的一般表达式，不仅证明了最佳逼近解的存在唯一性，而且给出了此解的具体表达式．     

斜对称双对角矩阵特征值反问题

讨论了关于斜对称双对角矩阵的特征值反问题．即：已知一个n阶斜对称双对角矩阵的特征值和两个n-1阶子矩阵的部分特征值，则可求得该矩阵．最后给出了数值例子．     

由谱数据构造实双反对称矩阵

讨论实完全反对称矩阵的一个特秆值反问题．研究了实完全反对称矩阵的一些特征性质，构造一个实反对称矩阵使其各阶顺序主子矩阵具有指定的特征值．证明了：给定满足一定分隔条件的两组数，存在一个实完全反对称矩阵，使其各阶中心主子矩阵具有相应的特征值．     

关于一个半正定矩阵的Khatri-Rao乘积的不等式的讨论

得到的一个矩阵乘积不等式及其逆向不等式．应用这些结果，把一个半正定矩阵Khatri-Rao乘积的不等式推广到实对称矩阵．并给出了它的逆向不等式及其等式条件．     

关于酉对称矩阵的QR分解及其算法

为了简化大型行(列)酉对称矩阵的QR分解,研究了行(列)酉对称矩阵的性质,获得了一些新的结果,给出了行(列)酉对称矩阵的QR分解的公式和快速算法,它们可极大地减少行(列)酉对称矩阵的QR分解的计算量与存储量,并且不会丧失数值精度.同时推广和丰富了邹红星等(2002)的研究内容,拓宽了实际应用领域的范围.     

一类亚半正定矩阵的左右逆特征值问题

1．引言在工程技术中常常遇到这样一类逆特征值问题：要求在一个矩阵集合S中，找具有给定的部分右特征对（特征值及相应的特征向量）和给定的部分左特征对（特征值及相应的特征向量）的矩阵．文［2］，［3］讨论了S为。x。实矩阵集合的情形．文［4］－［7］对S为nxn实对称矩阵．对称正定矩阵，对称半正定矩阵集合的情形进行了讨论．文【川讨论了S为亚正定阵集合的情形．并提到了对于亚半正定矩阵的情形目下无人涉及，有待进一步研究．本文将对S为nxn亚半正定矩阵集合的情形进行讨论．给出了亚半正定矩阵的左右逆特征值问题有解的充要条件…     

关于矩阵迹的平均不等式

关于矩阵迹的平均不等式黄礼平（湘潭矿业学院基础课部４１１２０１）近年来，对矩阵迹的不等式研究活跃，本文给出两个矩阵迹的平均不等式．定理１设Ａ，Ｂ，Ｃ均为ｎ阶半正定Ｈｅｒｍｉｔｅ矩阵，则特别，我们有推论１设Ａ，Ｂ，Ｃ均为ｎ阶正定实对称矩阵，则诸等号当且...     

实对称矩阵的一类逆特征值问题

利用矩阵的奇异值分解及广义逆，给出了矩阵约束下矩阵反问题AX=B有实对称解的充分必要条件及其通解的表达式．此外，给出了在矩阵方程的解集合中与给定矩阵的最佳逼近解的表达式．     

Symmetric Weighing Matrices Constructed using Group Matrices

A weighing matrix of order n and weight m² is a square matrix M of order n with entries from {-1,0,+1} such that MM^T=m²I where I is the identity matrix of order n. If M is a group matrix constructed using a group of order n, M is called a group weighing matrix. Recently, group weighing matrices were studied intensively, especially when the groups are cyclic and abelian. In this paper, we study the abelian group weighing matrices that are symmetric, i.e.M^T=M. Some new examples are found. Also we obtain a few exponent bounds on abelian groups that admit symmetric group weighing matrices. In particular, we prove that there is no symmetric abelian group weighing matrices of order 2p^r and weight p² where p is a prime and p≥ 5.Communicated by: K.T. Arasu     

An implicit QR algorithm for symmetric semiseparable matrices

The QR algorithm is one of the classical methods to compute the eigendecomposition of a matrix. If it is applied on a dense n × n matrix, this algorithm requires O(n³) operations per iteration step. To reduce this complexity for a symmetric matrix to O(n), the original matrix is first reduced to tridiagonal form using orthogonal similarity transformations. In the report (Report TW360, May 2003) a reduction from a symmetric matrix into a similar semiseparable one is described. In this paper a QR algorithm to compute the eigenvalues of semiseparable matrices is designed where each iteration step requires O(n) operations. Hence, combined with the reduction to semiseparable form, the eigenvalues of symmetric matrices can be computed via intermediate semiseparable matrices, instead of tridiagonal ones. The eigenvectors of the intermediate semiseparable matrix will be computed by applying inverse iteration to this matrix. This will be achieved by using an O(n) system solver, for semiseparable matrices. A combination of the previous steps leads to an algorithm for computing the eigenvalue decompositions of semiseparable matrices. Combined with the reduction of a symmetric matrix towards semiseparable form, this algorithm can also be used to calculate the eigenvalue decomposition of symmetric matrices. The presented algorithm has the same order of complexity as the tridiagonal approach, but has larger lower order terms. Numerical experiments illustrate the complexity and the numerical accuracy of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.     

A small note on the scaling of symmetric positive definite semiseparable matrices

In this paper we will adapt a known method for diagonal scaling of symmetric positive definite tridiagonal matrices towards the semiseparable case. Based on the fact that a symmetric, positive definite tridiagonal matrix satisfies property A, one can easily construct a diagonal matrix such that has the lowest condition number over all matrices , for any choice of diagonal matrix . Knowing that semiseparable matrices are the inverses of tridiagonal matrices, one can derive similar properties for semiseparable matrices. Here, we will construct the optimal diagonal scaling of a semiseparable matrix, based on a new inversion formula for semiseparable matrices. Some numerical experiments are performed. In a first experiment we compare the condition numbers of the semiseparable matrices before and after the scaling. In a second numerical experiment we compare the scalability of matrices coming from the reduction to semiseparable form and matrices coming from the reduction to tridiagonal form. *The research was partially supported by the Research Council K.U. Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann–Hilbert problems, random matrices and Padé–Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister's Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). The scientific responsibility rests with the authors. The second author participates in the SCCM program, Gates 2B, Stanford University, CA, USA and is also partially supported by the NSF. The first author visited the second one with a grant by the Fund for Scientific Research–Flanders (Belgium).     

A Lanczos‐type algorithm for the QR factorization of regular Cauchy matrices

We present a fast algorithm for computing the QR factorization of Cauchy matrices with real nodes. The algorithm works for almost any input matrix, does not require squaring the matrix, and fully exploits the displacement structure of Cauchy matrices. We prove that, if the determinant of a certain semiseparable matrix is non‐zero, a three term recurrence relation among the rows or columns of the factors exists. Copyright © 2002 John Wiley & Sons, Ltd.     

An Implicit Q Theorem for Hessenberg-like Matrices

The implicit Q theorem for Hessenberg matrices is a widespread and powerful theorem. It is used in the development of, for example, implicit QR algorithms to compute the eigendecomposition of Hessenberg matrices. Moreover it can also be used to prove the essential uniqueness of orthogonal similarity transformations of matrices to Hessenberg form. The theorem is also valid for symmetric tridiagonal matrices, proving thereby also in the symmetric case its power. Currently there is a growing interest to so-called semiseparable matrices. These matrices can be considered as the inverses of tridiagonal matrices. In a similar way, one can consider Hessenberg-like matrices as the inverses of Hessenberg matrices. In this paper, we formulate and prove an implicit Q theorem for the class of Hessenberg-like matrices. We introduce the notion of strongly unreduced Hessenberg-like matrices and also a method for transforming matrices via orthogonal transformations to this form is proposed. Moreover, as the theorem is valid for Hessenberg-like matrices it is also valid for symmetric semiseparable matrices. The research was partially supported by the Research Council K.U.Leuven, project OT/00/16 (SLAP: Structured Linear Algebra Package), by the Fund for Scientific Research–Flanders (Belgium), projects G.0078.01 (SMA: Structured Matrices and their Applications), G.0176.02 (ANCILA: Asymptotic aNalysis of the Convergence behavior of Iterative methods in numerical Linear Algebra), G.0184.02 (CORFU: Constructive study of Orthogonal Functions) and G.0455.0 (RHPH: Riemann-Hilbert problems, random matrices and Padé-Hermite approximation), and by the Belgian Programme on Interuniversity Poles of Attraction, initiated by the Belgian State, Prime Minister’s Office for Science, Technology and Culture, project IUAP V-22 (Dynamical Systems and Control: Computation, Identification & Modelling). This research was partially supported by by MIUR, grant number 2004015437 (third author). The scientific responsibility rests with the authors.     

Refined Enumerations of Some Symmetry Classes of Alternating-Sign Matrices

Using determinant representations for partition functions of the corresponding variants of square-ice models and the method recently proposed by one of us, we investigate refined enumerations of vertically symmetric alternating-sign matrices, off-diagonally symmetric alternating-sign matrices, and alternating-sign matrices with a U-turn boundary. For all these cases, we find explicit formulas for refined enumerations. In particular, we prove the Kutin–Yuen conjecture.     

Universality for the Conjugate Gradient and MINRES Algorithms on Sample Covariance Matrices

We present a probabilistic analysis of two Krylov subspace methods for solving linear systems. We prove a central limit theorem for norms of the residual vectors that are produced by the conjugate gradient and MINRES algorithms when applied to a wide class of sample covariance matrices satisfying some standard moment conditions. The proof involves establishing a four-moment theorem for the so-called spectral measure, implying, in particular, universality for the matrix produced by the Lanczos iteration. The central limit theorem then implies an almost-deterministic iteration count for the iterative methods in question. © 2022 Wiley Periodicals LLC.     

有理对称矩阵的准正交相似

对一个有理对称矩阵A,用线性代数的理论直接证明了存在准正交矩阵P使得P~(-1)AP成为A的有理标准形,并给出了P的算法.     

On tridiagonal matrices unitarily equivalent to normal matrices

In this article the unitary equivalence transformation of normal matrices to tridiagonal form is studied.It is well-known that any matrix is unitarily equivalent to a tridiagonal matrix. In case of a normal matrix the resulting tridiagonal inherits a strong relation between its super- and subdiagonal elements. The corresponding elements of the super- and subdiagonal will have the same absolute value.In this article some basic facts about a unitary equivalence transformation of an arbitrary matrix to tridiagonal form are firstly studied. Both an iterative reduction based on Krylov sequences as a direct tridiagonalization procedure via Householder transformations are reconsidered. This equivalence transformation is then applied to the normal case and equality of the absolute value between the super- and subdiagonals is proved. Self-adjointness of the resulting tridiagonal matrix with regard to a specific scalar product is proved. Properties when applying the reduction on symmetric, skew-symmetric, Hermitian, skew-Hermitian and unitary matrices and their relations with, e.g., complex symmetric and pseudo-symmetric matrices are presented.It is shown that the reduction can then be used to compute the singular value decomposition of normal matrices making use of the Takagi factorization. Finally some extra properties of the reduction as well as an efficient method for computing a unitary complex symmetric decomposition of a normal matrix are given.     

Hadamard Matrices Modulo p and Small Modular Hadamard Matrices

We use modular symmetric designs to study the existence of Hadamard matrices modulo certain primes. We solve the 7‐modular and 11‐modular versions of the Hadamard conjecture for all but a finite number of cases. In doing so, we state a conjectural sufficient condition for the existence of a p‐modular Hadamard matrix for all but finitely many cases. When 2 is a primitive root of a prime p, we conditionally solve this conjecture and therefore the p‐modular version of the Hadamard conjecture for all but finitely many cases when , and prove a weaker result for . Finally, we look at constraints on the existence of m‐modular Hadamard matrices when the size of the matrix is small compared to m.