矩阵特征多项式的图论计算公式

给出了赋权有向图邻接矩阵特征多项式的图论计算公式,从而得到了一般矩阵特征多项式的图论计算方法,并且研究了赋权有向图邻接矩阵特征多项式和谱半径的一些性质．     

On a certain canonical form

We amplify the well-known result due to Dlab and Ringel on the reduction of a real rectangular matrix to canonical form by formally complex transformations of rows and columns. Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 514–519, April, 2000.     

Reduction of matrices by means of equivalent transformations and similarity transformations

The structure of polynomial matrices in connection with their reducibility by semiscalar-equivalent transformations and similarity transformations to simpler forms is considered. In particular, the canonical form of polynomial matrices without multiple characteristic roots with respect to the above transformations is indicated. This allows one to establish a canonical form with respect to similarity for a certain type of finite collections of numerical matrices. Translated fromMatematicheskie Zametki, Vol. 64, No. 5, pp. 769–782, November, 1998.     

Solvability of matrix equations in rings of quasi-diagonal matrices and similarity of matrix polynomials

A certain standard form is found for a complex matrix with respect to equivalent transformations by quasi-diagonal matrices. The solvability of certain matrix equations in the rings of quasi-diagonal matrices is examined using this standard form.     

求矩阵到其Jordan标准形的过渡矩阵的新算法

本文给出了一种求矩阵到其Ｊｏｒｄａｎ标准形的过渡矩阵的新算法，与其它现有的算法相比，此算法最简单有效，且最易于编制程序以利用计算机计算．     

三阶矩阵的平方根

本文利用矩阵的特征值和Jordan标准形，给出了三阶矩阵的所有平方根。     

On tridiagonal conjugate-normal matrices

Conjugate-normal matrices play the same role in the theory of unitary congruences as conventional normal matrices do with respect to unitary similarities. Naturally, the properties of both matrix classes are fairly similar up to the distinction between the congruence and similarity. However, in certain respects, conjugate-normal matrices differ substantially from normal ones. Our goal in this paper is to indicate one of such distinctions. It is shown that none of the familiar characterizations of normal matrices having the irreducible tridiagonal form has a natural counterpart in the case of conjugate-normal matrices.     

矩阵多项式的平方根矩阵

研究了矩阵多项式的开平方问题,给出了矩阵多项式能开平方的充分必要条件及其平方根矩阵的个数,包含并推广了文[1]中的主要结论.     

用广义特征矩阵计算若当链

In this paper, we introduce a method to define generalized characteristic matrices of a defective matrix by the common form of Jordan chains. The generalized characteristic matrices can be obtained by solving a system of linear equations and they can be used to compute Jordan basis.     

Orthogonal similarity transformation of a symmetric matrix into a diagonal-plus-semiseparable one with free choice of the diagonal

In this paper we describe an orthogonal similarity transformation for transforming arbitrary symmetric matrices into a diagonal-plus-semiseparable matrix, where we can freely choose the diagonal. Very recently an algorithm was proposed for transforming arbitrary symmetric matrices into similar semiseparable ones. This reduction is strongly connected to the reduction to tridiagonal form. The class of semiseparable matrices can be considered as a subclass of the diagonalplus- semiseparable matrices. Therefore we can interpret the proposed algorithm here as an extension of the reduction to semiseparable form. A numerical experiment is performed comparing thereby the accuracy of this reduction algorithm with respect to the accuracy of the traditional reduction to tridiagonal form, and the reduction to semiseparable form. The experiment indicates that all three reduction algorithms are equally accurate. Moreover it is shown in the experiments that asymptotically all the three approaches have the same complexity, i.e. that they have the same factor preceding the n³ term in the computational complexity. Finally we illustrate that special choices of the diagonal create a specific convergence behavior. The research was partially supported by the Research Council K.U.Leuven, project OT/05/40 (Large rank structured matrix computations), 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.     

正交矩阵的特征多项式及特征根

以《高等代数习题解》(杨子胥)的两道习题为理论根据,应用正交矩阵的若干性质,给出了正交矩阵特征多项式系数的规律.     

矩阵代数上的局部合同变换和局部相似变换

令F表示任意域,Mn（F）表示由F上所有n×n矩阵形成的结合代数.本文的目的是研究Mn（F）上具有如下性质的两类线性映射,其中一类线性映射在Mn（F）上每一点的取值与Mn（F）的某个合同变换在该点的取值相同,另一类线性映射在Mn（F）上每一点的取值与Mn（F）的某个相似变换在该点的取值相同,随着Mn（F）上的点不同,这些合同变换和相似变换可能也不同.利用矩阵的秩、幂等阵以及幂零阵的性质,通过矩阵计算的方法证明了第一类线性映射或者是合同变换或者是合同变换与转置变换的复合,第二类线性映射或者是相似变换或者是相似变换与转置变换的复合.由这个结果可知存在真正意义上的局部合同变换和局部相似变换,从而丰富了局部映射理论的研究。     

Jordan canonical form of Pascal-type matrices via sequences of binomial type

In this paper, we study the Jordan canonical form of the generalized Pascal functional matrix associated with a sequence of binomial type, and demonstrate that the transition matrix between the generalized Pascal functional matrix and its Jordan canonical form is the iteration matrix associated with the binomial sequence. In addition, some combinatorial identities are derived from the corresponding matrix factorization.     

关于幂等矩阵秩的一个命题的证明和推广

给出秩命题"n阶方阵A为幂等矩阵等价于r(A)+r(E-A)=n"的五种证明,并推广其结论,从而刻画了几类矩阵的秩特征(见定理1-3).     

Orthogonal similarity transformation into block‐semiseparable matrices of semiseparability rank k

Very recently, an algorithm, which reduces any symmetric matrix into a semiseparable one of semi‐ separability rank 1 by similar orthogonality transformations, has been proposed by Vandebril, Van Barel and Mastronardi. Partial execution of this algorithm computes a semiseparable matrix whose eigenvalues are the Ritz‐values obtained by the Lanczos' process applied to the original matrix. Also a kind of nested subspace iteration is performed at each step. In this paper, we generalize the above results and propose an algorithm to reduce any symmetric matrix into a similar block‐semiseparable one of semiseparability rank k, with k ∈ ?, by orthogonal similarity transformations. Also in this case partial execution of the algorithm computes a block‐semiseparable matrix whose eigenvalues are the Ritz‐values obtained by the block‐Lanczos' process with k starting vectors, applied to the original matrix. Subspace iteration is performed at each step as well. Copyright © 2005 John Wiley & Sons, Ltd.     

Universal similarity factorization equalities for commutative quaternions and their matrices

In this work, we have established universal similarity factorization equalities over the commutative quaternions and their matrices. Based on these equalities, real matrix representations of commutative quaternions and their matrices have been derived, and their algebraic properties and fundamental equations have been determined. Moreover, illustrative examples are provided to support our results.     

On the divisor function of sets with even partition functions

Summary For P∈ F₂[z] with P(0)=1 and deg(P)≧ 1, let A =A(P) be the unique subset of N (cf. [9]) such that Σ_n≧0 p(A,n)zⁿ ≡ P(z) mod 2, where p(A,n) is the number of partitions of n with parts in A. To determine the elements of the set A, it is important to consider the sequence σ(A,n) = Σ _{d|n, d∈A} d, namely, the periodicity of the sequences (σ(A,2^kn) mod 2^k+1)_n≧1 for all k ≧ 0 which was proved in [3]. In this paper, the values of such sequences will be given in terms of orbits. Moreover, a formula to σ(A,2^kn) mod 2^k+1 will be established, from which it will be shown that the weight σ(A₁,2^kz_i) mod 2^k+1 on the orbit <InlineEquation ID=IE"1"><EquationSource Format="TEX"><![CDATA[<InlineEquation ID=IE"2"><EquationSource Format="TEX"><![CDATA[$]]></EquationSource></InlineEquation>]]></EquationSource></InlineEquation>z_i$ is moved on some other orbit z_j when A₁ is replaced by A₂ with A₁= A(P₁) and A₂= A(P₂) P₁ and P₂ being irreducible in F₂[z] of the same odd order.     

On the multiplicity of the eigenvalues of a graph

Given a graph G with characteristic polynomial ϕ(t), we consider the ML-decomposition ϕ(t) = q ₁(t)q ₂(t)² ... q _m (t)^m, where each q _i (t) is an integral polynomial and the roots of ϕ(t) with multiplicity j are exactly the roots of q _j (t). We give an algorithm to construct the polynomials q _i (t) and describe some relations of their coefficients with other combinatorial invariants of G. In particular, we get new bounds for the energy E(G) = |λ_i| of G, where λ₁, λ₂, ..., λ_n are the eigenvalues of G (with multiplicity). Most of the results are proved for the more general situation of a Hermitian matrix whose characteristic polynomial has integral coefficients. This work was done during a visit of the second named author to UNAM.     

On the eigenvalues of specially low-rank perturbed matrices

We study the eigenvalues of a matrix A perturbed by a few special low-rank matrices. The perturbation is constructed from certain basis vectors of an invariant subspace of A, such as eigenvectors, Jordan vectors, or Schur vectors. We show that most of the eigenvalues of the low-rank perturbed matrix stayed unchanged from the eigenvalues of A; the perturbation can only change the eigenvalues of A that are related to the invariant subspace. Existing results mostly studied using eigenvectors with full column rank for perturbations, we generalize the results to more general settings. Applications of our results to a few interesting problems including the Google’s second eigenvalue problem are presented.     

Backlund Transformations for the Isospectral and Non-isospectral Matrix Kdv Hierarchies

By transforming the usual Lax pairs of isospectral and non-isospectral matrix Kdv hierarchies into Lax pairs Riccati form, a unified explicit from of Backlund transformations and superposition formulas for these two kinds of hierarchies of equations can be ohtaincd.